arXiv:1505.07493v1 [cs.IT] 27 May 2015
Dualilty Preserving Gray Maps: (Pseudo) Self-dual Bases and Symmetric Bases Steve Szabo ∗ Department of Mathematics and Statistics Eastern Kentucky University Richmond, KY 40475 Felix Ulmer† IRMAR, CNRS, UMR 6625 Universit´e de Rennes 1 Universit´e europ´eenne de Bretagne Campus de Beaulieu, F-35042 Rennes
Abstract Given a finite ring A which is a free left module over a subring R of A, two types of R-bases are defined which in turn are used to define duality preserving maps from codes over A to codes over R. The first type, pseudo-self-dual bases, are a generalization of trace orthogonal bases for fields. The second are called symmetric bases. Both types are illustrated with skew cyclic codes which are codes that are A-submodules of the skew polynomial ring A[X; θ]/hX n − 1i (the classical cyclic codes are the case when θ = id). When A is commutative, there exists criteria for a skew cyclic code over A to be self-dual. With this criteria and a duality preserving map, many self-dual codes over the subring R can easily be found. In this fashion, numerous examples are given, some of which are not chain or serial rings.
Keywords: Codes over Rings, Self-Dual Codes, Trace Orthogonal Basis, Symmetric Basis, Codes over Noncommutative Rings. MSC2010: 94B05, 94B60
1
Introduction and Overview
Codes over rings have been a major topic in coding theory ever since the discovery of the connection between linear codes over Z4 to some good non-linear ∗ e-mail: † e-mail:
[email protected] [email protected]
1
binary codes in [14]. They showed that there is a (Gray) map from codes over Z4 to codes over F22 for which the mentioned good linear codes were the images of linear Z4 codes. Many classical constructions of codes over fields like cyclic codes and geometric codes have been generalized to rings ([5, 21]). The idea of mapping codes from larger rings onto codes over smaller rings has now been a growing topic where not only non-linear but also linear mappings are used. Example 1.1. Consider the F2 -algebra A = F2 [x]/(x2 ) (see [16]). Using the ordered F2 basis (1, x) of A, any n-length code over A can be mapped to a 2n-length code over F2 via the bijection Φ : An → (F2 )2n ; (a1 , a2 , . . . , an ) → (α1,1 , α1,2 , α2,1 , α2,2 , . . . , αn,1 , αn,2 ) where ai = αi,1 + αi,2 x is the representation of ai in the basis (1, x). The image of a linear code over A is a code over F2 . Specifically, consider g = X 2 + xX + 1 ∈ A[X]. This polynomial is a divisor of X 4 − 1 ∈ A[X] and generates a cyclic code (g)/(X 4 − 1) ⊂ A[X]/(X 4 − 1) of length 4 over A. In the standard correspondence of (g)/(X 4 − 1) with A4 , g corresponds to (1, x, 1, 0). Mapping g to F82 using the basis (1, x), we have (1, 0, 0, 1, 1, 0, 0, 0). Also, the code word x · (X 2 + xX + 1) is mapped to (0, 1, 0, 0, 0, 1, 0, 0). Applying this argument to the code word X · g we see that 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1
is a generator matrix of the binary image of the cyclic code over A generated by g. Self-dual codes have become a central topic in coding theory due to their connections to other fields of mathematics such as block designs [8]. In many instances, self-dual codes have been found by first finding a code over a ring and then mapping this code onto a code over a subring through a map that preserves duality. Often, these maps have been found through ad hoc methods. For instance, in [25], the local Frobenius non-chain rings of order 16 were found and a map that preserves duality was presented for each ring. In the literature, the mappings typically map to codes over F2 , F4 and Z4 since codes over these rings have had the most use. Example 1.2. We consider again the cyclic code C = (g)/(X 4 −1) ⊂ A[X]/(X 4 − 1) of length 4 over A = F2 [x]/(x2 ) generated by g(X) = X 2 + xX + 1 ∈ A[X]. Since g equals its reciprocal polynomial X 2 · g(1/X), the code C is self-dual of length 4 over A. However, the binary image of C obtained in previous example using the basis (1, x) is not a self-dual binary code. If we use the F2 -basis (1, x + 1) to map to F82 , then the image of C is the type II binary code [8, 4, 4]2
2
generated by
1 0 0 0
0 1 0 0
1 1 1 0
1 1 0 1
1 0 1 1
0 1 1 1
0 0 1 0
0 0 . 0 1
Note that the code word (1, x, 1, 0) corresponding to g ∈ A[X]/(X 4 − 1) can be written (1 · 1 + 0 · (x + 1), 1 · 1 + 1 · (x + 1), 1 · 1 + 0 · (x + 1), 0 · 1 + 0 · (x + 1)) and maps to the first row (1, 0, 1, 1, 1, 0, 0, 0), while the second row is the image of (1 + x) · g. It is well known ([16]) that in the F2 -basis (1, x + 1) of A, the image of a self-dual code over A is always a self-dual code over F2 . Therefore, the basis (1, x + 1) preserves duality under this mapping. We aim to give criteria for bases which guarantees the basis preserves duality. Our setting is as described above: A is a finite ring that is a free left module over a subring R of A. We present two types of bases that have this duality preserving property. The first, pseudo-self-dual bases, are a generalization of trace orthogonal bases which have been defined for algebras over finite fields in [28] and [6]. The second, symmetric bases, are a generalization of the same for finite fields defined in [27]. In the case of symmetric bases, it is required that A, not only be a left R-module, but a left R-algebra which in turn then requires that R ⊂ Z(A). Other than in [3], where a criterion is given to find a basis for Fq [x]/(xt ) that maps self-dual codes over Fq [x]/(xt ) to self-dual codes over Fq , the authors are unaware of general methods for finding duality preserving maps. The methods herein are much more general than that in [3]. It is shown here that the criteria for an Fq -basis for Fq [x]/(xt ) to preserve duality given in [3] is equivalent to the basis being symmetric (see Proposition 3.19). Given a ring A and a subring R for which one has a duality preserving map as described along with a method for finding self-dual codes over A ([9, 17]), ultimately one can find self-dual codes over R. In order to illustrate the use of duality preserving maps in this fashion, we will discuss codes over a ring A for which the dual of a code (and therefore self-dual codes) can be constructed algebraically. The class of codes under consideration are the well known skew-cyclic codes over A which are left ideals of A[X; θ]/(X n − 1). Here A[X; θ] is the skew polynomial ring where θ is an automorphism of A. These codes have the property that if (a0 , . . . , an−2 , an−1 ) is in a code C then (θ(an−1 ), θ(a0 ), . . . , θ(an−2 )) is in C. Notice that if θ is the identity, these codes are simply the classical cyclic codes over A. In many cases the ring A[X; θ] is not a unique factorization domain, which leads to many different skew-cyclic codes and self-dual skew-cyclic codes. The paper is organized as follows: Section 2 contains preliminaries and definitions needed throughout the rest of the paper. Section 3 lays out the two methods for finding duality preserving bases. In Section 4, skew polynomial rings and skew cyclic codes are defined and a few illustrative examples are given. Finally, in Section 5, many self-dual skew cyclic codes are found along with the parameters of their images under a duality preserving map. 3
2
Preliminaries
For a ring A, Aut(A) is the automorphism group of A, a subset C of An is an n length code over A and if C is an A-submodule of An then C is an A-linear code over A. For a finite ring A and a subgroup H of Aut(A), the fixed subring of H is AH = {a ∈ A|h(a) = a for all h ∈ H} and the trace function with respect to H is X T rH : A → A; a 7→ h(a). h∈H
Lemma 2.1. Let A be a ring and H be a subgroup of Aut(A). The trace function with respect to H is both a left and right AH -linear map and for a ∈ A, T rH (a) ∈ AH . Proof. Let a ∈ A, b ∈ AH and g ∈ H. Then T rH (ba) =
X
h(ba) =
h∈H
X
h(b)h(a) =
h∈H
X
bh(a) = b
h∈H
X
!
h(a) .
h∈H
So, T rH is left AH -linear. Similarly, T rH is right AH -linear. Also, ! X X X g(T r(a)) = g h(a) = g(h(a)) = h(a) = T r(a) h∈H
h∈H
h∈H
showing T rH (a) ∈ AH . For a ring A, an anti-automorphism σ on A and a left A-module M , a σsesquilinear form on M is a map h·, ·i : M × M → A such that if x, y, z ∈ M and a ∈ A then hx + z, yi = hx, yi + hz, yi, hx, y + zi = hx, yi + hx, zi, hax, yi = ahx, yi and hx, ayi = hx, yiσ(a). In addition, if σ(hx, yi) = hy, xi then the form is called a σ-hermitian form. A σ-sesquilinear form with the property that hx, yi = 0 ⇐⇒ hy, xi = 0 is called reflexive. Clearly a σ-hermitian form is reflexive. Proposition 2.2. Let A be a ring, σ be an anti-automorphism on A and M be a left A-module. Define the map h·, ·i : Ak × Ak → A; hx, yi =
k X
xi σ(yi ).
i=1
Then h·, ·i is a σ-sesquilinear form. Furthermore, h·, ·i is a σ-hermitian form if and only if σ is involutory. Proof. Clearly, h·, ·i is a σ-sesquilinear form. Assume h·, ·i is hermitian. For a ∈ A, a = ah(1, 0, . . . , 0), (1, 0, . . . , 0)i = h(a, 0, . . . , 0), (1, 0, . . . , 0)i. Since σ is hermitian, σ 2 (a) = σ 2 (h(a, 0, . . . , 0), (1, 0, . . . , 0)i) = a showing σ is involutory. Pk Now, assume σ is involutory. For x, y ∈ Ak , σ(hx, yi) = σ( i=1 xi σ(yi )) = Pk Pk i=1 σ(xi σ(yi )) = i=1 yi σ(xi ) = hy, xi showing h·, ·i is σ-hermitian. 4
When the form in Proposition 2.2 is hermitian, it is known as the standard σ-hermitian form on Ak which we denote as hx, yiAk . If σ is the identity map, hx, yiAk is known as the the standard bilinear form on Ak . Since the identity map is an involution if and only if A is commutative, we may only consider the standard bilinear form on Ak when A is commutative. If R is a subring of A such that σ(R) = R then σ|R is an involution on R. Then, if entries in Ak are restricted to Rk , the standard σ-hermitian form on Ak is the standard σ|R -hermitian form on Rk . We will specialize to the standard σ-hermitian form on Ak for which we define the orthogonal of a code. From Proposition 2.2, we know then that σ is an involution (an involutory anti-automorphism). For a finite ring A, the standard σ-hermitian form on Ak and an A-linear code C ⊂ Ak , the dual code of C, denoted by C ⊥ , is C ⊥ = {v | hv, ciAk = 0, ∀c ∈ C}. A code is self orthogonal if C ⊂ C ⊥ and self dual if C = C ⊥ . The dual we have defined is typically referred to as the left dual. There is an analog notion of a right dual. These of course when working over a commutative ring are identical. Furthermore, since we have defined the dual based on a hermitian form and a hermitian form is reflexive (see definition above), the left and right duals coincide over non-commutative rings as well, that is {v | hv, ciAk = 0, ∀c ∈ C} = {v | hc, viAk = 0, ∀c ∈ C}. From Wood’s fundamental results in [29] we have the following. Lemma 2.3. Let A be a finite Frobenius ring and C be a linear code over A. Then |C||C ⊥ | = |A|n . For this and other reasons presented in [29], most coding theorists restrict there study to codes over Frobenius rings. For the definition and details about Frobenius rings see [29]. We will keep things general and not restrict the development of the theory to Frobenius until necessary. To illustrate one of the difficulties of working on codes over non-Frobenius rings we provide the following example. Example 2.4. Let A = huF22,v[u,v] 2 ,uvi and let C = (u, v) ⊳ A which is an A-linear ⊥ code of length 1. Then C = C. Now, |C| = 4. Then |C||C ⊥ | = 16 > 8 = |A|. We see that A is non-Frobenius since the Jacobson radical J(A) = hu, vi is not isomorphic to the socle of A, Soc(R) = hui + hu + vi + hvi as A modules.
3
Duality Preserving Bases
Throughout this section let n ∈ N, let A be a finite unitary ring which is a free left module over a unitary subring R of A, B = (v1 , . . . , vr ) be an ordered left R-basis for A and σ be an involution on A such that σ(R) = R. Define the component maps ρ : A → Rr ; a1 v1 + · · · + ar vr 7→ (a1 , . . . , ar ) 5
(3.1)
and Φ : An → Rrn ; (α1 , . . . , αn ) 7→ (ρ(α1 ), . . . , ρ(αn )). We first show the connection between the inner product on An and the inner product on Rrn . To that end we introduce the following, let v1 σ(v1 ) . . . v1 σ(vr ) T .. .. .. M = B σ(B) = , . . . vr σ(v1 ) . . .
vr σ(vr )
and let M be the block diagonal nr × nr matrix with M on the diagonal.
Lemma 3.1. Let x, y ∈ An . Then hx, yiAn = Φ(x)Mσ(Φ(y))T . Proof. Let xi and yi be the i-th component of x and y respectively. Note that T for a ∈ A, a = ρ(a)B . Then hx, yiAn
=
n X
xi σ(yi ) =
=
T
T
ρ(xi )B σ(ρ(yi )B )
i=1
i=1
n X
n X
T
T
ρ(xi )B σ(B)σ(ρ(yi )) =
n X
T
ρ(xi )M σ(ρ(yi )) .
i=1
i=1
T
= Φ(x)Mσ(Φ(y))
In this section we investigate the sufficient conditions on B so that hx, yiAn = 0 implies hΦ(x), Φ(y)iRnr = 0 giving a so called duality preserving basis. We will give two separate conditions on B that guarantee it will preserve duality.
3.1
Pseudo-Self-dual Bases
In this subsection we consider pseudo-self-dual bases and show that such bases preserve duality. In [28], trace orthogonal bases for finite field extensions were defined. We extend this definition to include ring extensions in our setting as we consider the extension A ⊃ R. Definition 3.2. For a subgroup H of Aut(A) we define the following. B is a σtrace orthogonal basis with respect to H if for 1 ≤ i, j ≤ r, T rH (vi σ(vj )) = 0 if and only if i 6= j. In addition, if there exists γ ∈ A that is not a zero divisor, commutes with elements of R and T rH (vi σ(vi )) = γ then B is called a σpseudo-self-dual basis with respect to H. Furthermore, if γ = 1, B is called a σ-self-dual basis with respect to H. Example 3.3. The ring M2 (F3 ) has an automorphism group isomorphic to S4 , 10 involutions, a unique subfield F3 of order 3 and three subfields F9,1 , F9,2 andF9,3 of order 9 whose groups of units are generated respectively by 0 1 0 1 1 1 , and . Each subfield of order 9 is fixed by a 1 1 1 2 2 1 different subgroup H4,1 , H4,2 and H4,3 of order 4 of Aut(A). 6
1. For some involutions such as a σ: c
b d
7→
a 2c 2b d
,
M3 (F3 ) has σ-pseudo-self-dual bases only for F9,3 of the 3 subfields of order 9. 2. For some involutions such as a b 2a + 2b + 2c + 2d 2a + 2b + c + d σ: 7→ , c d 2a + b + 2c + d 2a + b + c + 2d M3 (F3 ) has σ-pseudo-self-dual bases for all 3 subfields of order 9. M3 (F3 ) has (a) 8 σ-self-dual and 56 F9,1 -bases with respect to σ-pseudo-self-dual 2 0 1 2 H4,1 (for example, , σ-self-dual), 1 1 0 2 (b) 48 σ-self-dual and 48 σ-pseudo-self-dual F9,2 -bases with respect to 1 2 1 0 H4,2 (for example , is σ-self-dual) and 2 1 2 0 (c) 8 σ-self-dual and 56 σ-pseudo-self-dual F9,3 -bases with respect to 1 2 2 0 is σ-self-dual). , H4,3 (for example 0 2 1 1 Lemma 3.4. Let H be a subgroup of the automorphism group of A such that R ⊂ AH . Assume B is a σ-trace w.r.tPH. Then for x = (x1 , . . . , xn ), y = Porthogonal r r (y1 , . . . , yn ) ∈ An where xi = j=1 αij vj , yi = k=1 βik vk ∈ A T rH (hx, yiAn ) =
n X r X
αij T rH (vj σ (vj )) σ (βij ) .
i=1 j=1
Proof. Since R ⊂ AH and B is σ-trace orthogonal w.r.t H, by Lemma 2.1 we have that ! r n r X X X αij vj σ βik vk T rH (hx, yiAn ) = T rH =
= =
i=1
j=1
i=1
j=1
k=1
! r r n X X X σ (vk ) σ (βik ) αij vj T rH
r r X n X X
k=1
αij T rH (vj σ (vk )) σ (βik )
i=1 j=1 k=1 n X r X
αij T rH (vj σ (vj )) σ (βij )
i=1 j=1
7
The next lemma shows that a map defined using a σ-pseudo-self-dual basis preserves orthogonality. It is important to note here that any σ hermitian form on A restricts to a hermitian form on R since σ(R) = R. Lemma 3.5. Let H be a subgroup of the automorphism group of A such that R ⊂ AH and x, y ∈ An . Assume B is σ-pseudo-self-dual basis w.r.t H and that hx, yiAn = 0. Then hΦ(x), Φ(y)iRnr = 0. Proof. Since B is σ-pseudo-self-dual basis w.r.t H there exists a γ ∈ A that commutes with R, is not a zero divisor and for 1 ≤ i, j ≤ r with i 6= j, T rH (vi σ(vi )) = γ and T rH (vi σ(vj )) = 0. By Lemma 3.4 and using definitions for x and y from that lemma we have 0 = T rH (0) = T rH (hx, yiAn ) =
n X r X
αij T rH (vj σ (vj )) σ (βij )
i=1 j=1
=
n X r X
αij γσ (βij ) = γ
n X r X
αij σ (βij ) = γhΦ(x), Φ(y)iRnr .
i=1 j=1
i=1 j=1
Since γ is not a zero divisor, hΦ(x), Φ(y)iRnr = 0. For an alternate view of the previous result, we return to the notation introduced at the beginning of this section. From Lemma 3.1 we have that T
hx, yiAn = Φ(x)Mσ(Φ(y)) . In the setting of Lemma 3.5 we have then 0 = =
T
T rH (0) = T rH (hx, yiAn ) = T rH (Φ(x)Mσ(Φ(y) )) T
T
Φ(x)T rH (M)σ(Φ(y) ) = Φ(x)(γInr σ(Φ(y) ) = γhΦ(x), Φ(y)iRnr .
Again, since γ is not a zero divisor, hΦ(x), Φ(y)iRnr = 0. The point here is this. In general, if T rH (M) = γInr which boils down to T rH (M ) = γIr , then B is a σ-pseudo-self-dual basis for A over R. If in addition, R ⊂ AH we see that B will preserve duality. Theorem 3.6. Let H be a subgroup of the automorphism group of A such that R ⊂ AH and let C be an n length linear code over A. Assume A and R are Frobenius rings and B is σ-pseudo-self-dual basis w.r.t H. Then Φ(C ⊥ ) = Φ(C)⊥ . Proof. By Lemma 3.5, since B is σ-pseudo-self-dual basis w.r.t H, Φ(C ⊥ ) ⊂ Φ(C)⊥ . We know |C| = |Φ(C)|. Since A and R are Frobenius rings, from Lemma 2.3, we have that |C ⊥ | =
|A|n |R|rn = = |Φ(C)⊥ |. |C| |Φ(C)|
8
Example 3.7. (see section 4 in [18]) For A = GR(4, 2), the Galois ring consists of all elements of the form α0 + α1 ξ where αi ∈ Z4 and ξ 2 + ξ + 1 = 0, the automorphism group is of order 2 generated by θ : A → A defined by θ : ξ 7→ 3ξ + 3. With σ = id or σ = θ, it turns out that A has no σ-self-dual basis over its prime ring Z4 , but has 8 σ-pseudo-self-dual bases over Z4 : {3ξ + 2, ξ + 1}, {ξ + 1, ξ + 2}, {3ξ + 2, 3ξ + 3}, {3ξ + 1, ξ} {3ξ + 3, ξ + 2}, {ξ, ξ + 3}, {3ξ, ξ + 3}, {3ξ, 3ξ + 1} For each basis and each involution, T rH (vσ(v)) = 3 for each basis element v. The basis {ξ, ξ + 3} was used in [9] to map Euclidean self-dual codes over A to Euclidean self-dual codes over Z4 . Example 3.8. Let A = M2 (F2 ). In [4], it was shown that codes over A can be mapped to codes over F4 . In [2] self-dual cyclic codes over A were studied. In the process they show (Proposition 2 of [2]) that there is a map from An to F2n 4 that preserves self-orthogonality for a particular hermitian form. It turns out that this map is defined using a σ-self-dual F4 basis for A and the hermitian form is the standard σ-hermitian form on A where σ is the anti-transpose on A. Much more can be said about this map, specifically that it preserves duality for not only the hermitian form based on the anti-transpose on A but also the hermitian form based on the transpose on A and a few others. This is what we shall show here. The following identifications for the elements of M2 (F2 ) will be used throughout. 1 0 0 0 0 1 1 1 I= , z= , i= , a= 0 1 0 0 1 0 1 1 1 0 0 1 0 0 0 0 e1 = , e2 = , e3 = , e4 = 0 0 0 0 1 0 0 1 0 1 1 0 1 1 1 1 u1 = , u2 = , u3 = , u4 = 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 t= , b= , l= , r= 0 0 1 1 1 0 0 1 We first compute the automorphisms and anti-automorphisms of A. The set of units of R is {I, i, u1 , u2 , u3 , u4 }. The units i, u2 and u3 are of multiplicative order 2 and the units u1 and u4 are of multiplicative order 3. The set {I, i, u1 , u2 } is an F2 -basis for A. Replacing i or u2 with u3 will still be a basis, as will replacing u1 with u4 . Now, any automorphism or anti-automorphism of A must send units to units of the same order. There are 12 such maps A → A that send units to units of the same order. They form a group isomorphic to a subgroup of S6 . Consider three of these maps which we express in cycle notation on the set of units. Let τ be the map (u2 u3 ), ψ be the map(u1 u5 ) and θ be the
9
map (iu2 u3 ). It turns out that τ is the transpose, ψ is the anti-transpose (the reflection of a matrix about the anti-diagonal) and a b b+d a+b+c+d θ: 7→ . c d b a+b Note that τ and ψ are anti-automorphisms on A and θ is an automorphism on A. From this we deduce that the automorphism group is isomorphic to S3 and generated by τ ψ and θ which is {id, θ, θ2 , τ ψ, τ ψθ, τ ψθ2 }. The set of antiisomorphisms are {τ, ψ, τ θ, ψθ, τ θ2 , ψθ2 }. The subset of these, {τ, ψ, τ θ, τ θ2 }, is the set of involutions of A, the remaining anti-isomorphisms having order 6. Let R = {I, z, u1, u4 }. Notice, R is a subring of A isomorphic to F4 and A is a free left R-module with left R-basis B = (I, i). Notice ψ is simply the frobenius map when restricted to R and is the conjugation map on A from Section 4.2 in [2]. Of course ψ(R) = R. Let H be the group generated by θ whose fixed ring is R. Since T rH (Iψ(i)) = T rH (iψ(I)) = z and T rH (Iψ(I)) = T rH (iψ(i)) = I, B is a ψ-self-dual basis w.r.t. H. Since A and R are Frobenius, by Theorem 3.6, the dual of a code over A is mapped to the dual of the image of the code over R where the hermitian form being considered is based on the anti-transpose. Since ψ restricted to R is simply the Frobenius map on F4 we are considering standard ψ-hermitian form on R. It can be similarly shown that B is a τ -self-dual basis w.r.t. H. The difference here is that τ is the identity on R. So, the τ -hermitian form restricted to R is simply standard Euclidean form. Using Magma, we found all σ-pseudo-self-dual F4 -bases for M2 (F2 ) for each given involution, σ. 1. These are the σ-pseudo-self-dual F4 -bases w.r.t. H = hθi for M2 (F2 ) for each involution σ: • σ = ψ: {I, i}, {I, u2}, {I, u3 }, {i, u1}, {i, u4}, {u1 , u2 }, {u1 , u3 }, {u2 , u4 }, {u3 , u4 } • σ = τ : {I, i}, {e1, e2 }, {l, r}, {e3, e4 }, {u1, u2 }, {u3 , u4 }. • σ = τ θ: {I, u2 }, {i, t}, {u1, u3 }, {r, a}, {b, e4}, {e2 , t} • σ = τ θ2 : {I, i}, {u1, u2 }, {u3, u4 }, {e1 , e2 }, {e3, e4 } 2. The following are the σ-pseudo-self-dual F2 -bases w.r.t. H = Aut(A) for A for each involution σ : • σ = ψ: none σ = τ : {u1 , u2 , u3 , u4 }, {e1 , e2 , e3 , e4 } • σ = τ θ : {i, u1, u3 , u4 }, {r, a, e2, t} • σ = τ θ2 : {i, u1 , u2 , u4 }, {r, a, b, e3 } Lemma 3.9. Let H be a subgroup of Aut(A) such that R ⊂ AH and ϕ be an involution on A such that ϕ(R) = R. Then 1. σ b = ϕσϕ is an involution on A such that σ b(R) = R 10
b = ϕHϕ is a subgroup of Aut(A) 2. H
3. Bb = {ϕσ(v)|v ∈ B} is an R-basis for A
b T r b (b 4. For vb, w b ∈ B, b = ϕ (T rH (wσ(v))). H v σ(w))
b Proof. The results 1-3 are straight forward. For 4, let vb, w b ∈ B. X X T rHb (b v σ(w)) b = h(b vσ b(w) b = ϕhϕ(ϕσ(v)ϕσϕ(ϕσ(w)) h∈H
b h∈H
= ϕ
X
h(wσ(v)) = ϕ (T rH (wσ(v)))
h∈H
The next proposition is immediate from Lemma 3.9. Proposition 3.10. Let H be a subgroup of the automorphism group of A such that R ⊂ AH , ϕ be an involution on A such that ϕ(R) = R. Then B is σ-pseudoself-dual basis w.r.t H if and only if {ϕσ(v)|v ∈ B} is a ϕσϕ-pseudo-self-dual basis w.r.t ϕHϕ over R of A. Example 3.11. We return to Example 3.8. Since ψ conjugated with any other involution is ψ, applying Proposition 3.10 to a ψ-pseudo-self-dual basis will produce a ψ-pseudo-self-dual basis. Similarly, τ conjugated τ or ψ is τ , applying Proposition 3.10 in these cases will only produce a τ -pseudo-self-dual basis from τ -pseudo-self-dual basis. More interestingly though, (τ θ)τ (τ θ) = τ θ2 . Now if we apply Proposition 3.10 with the involution τ θ to a τ -pseudo-self-dual basis, we obtain a τ θ2 -pseudo-self-dual basis. Similarly, (τ θ2 )τ (τ θ2 ) = τ θ. If we apply Proposition 3.10 with the involution τ θ2 to a τ -pseudo-self-dual basis, we obtain a τ θ-pseudo-self-dual basis. This explains why there are the same number of τ , τ θ and τ θ2 pseudo-self-dual bases. It is not always the case that A has a σ-pseudo-self-dual basis over R. For instance, from the full list of indecomposable commutative rings of order 16 given in [25], none of the rings F4 [x]/(x2 ), F2 [x]/(x2 + y 2 , xy), F2 [x]/(x2 , y 2 ), Z4 [x]/(x2 + 2x), Z4 [x]/(x2 + 2), Z4 [x]/(x2 + 2x + 2) and Z4 [x]/(x2 ) have a σpseudo-self-dual basis for any proper subring. This is not to say that no duality preserving map exists over these rings as will be seen in the next subsection where we look at an alternative property which guarantees duality preservation. In the case of Z4 [x]/(x2 + 2x) it is true that no such duality preserving map exists as this was shown in [26]. Similarly, we can find examples where A is noncommutative and does not have a σ-pseudo-self-dual basis over R. For 4 [x;θ] instance, F4hx[x;θ] where θ is the Frobenius map on F4 extended to Fhx 2i 2i .
11
3.2
Symmetric Bases
In this section assume additionally that A is a free R-algebra not simply a left R-module. Although we do not assume A is commutative here, if we assume B is symmetric (defined below), A must be commutative. See Lemma 3.16. So, this section in reality is strictly about commutative alphabets. In the commutative case, σ is simply an order two automorphism or the identity map. This allows us to consider the Euclidean inner product on commutative rings in our setting which is not possible in the non-commutative case as the identity map is not an involution in that case. Definition 3.12. For a ∈ A, let Ma denote the matrix w.r.t. B representing the linear transformation of right multiplication by a, i.e. ρ(v1 a) .. Ma = . ρ(vr a)
In the above definition Ma acts on row vectors from Rr on the right. It is well know that sending a ∈ A to Ma is an embedding of A in Mr (R). Since for a ∈ A, ρ(a) is just the representation of a in B, the following lemma is immediate. Lemma 3.13. Let a, b ∈ A. Then ρ(ab) = ρ(a)Mb and ab = ρ−1 (ρ(a)Mb ).
In [27], symmetric bases were defined for field extensions. As we did with trace orthogonal bases in the last section, we extend the definition of symmetric bases to include the ring extensions we are considering. T
Definition 3.14. B is a symmetric basis if for any v ∈ B, Mv = Mv . If B is symmetric, since {Mv |v ∈ B} is a basis for {Ma |a ∈ A} we have the following lemma. T
Lemma 3.15. Assume B is symmetric. Then for a ∈ A, Ma = Ma . If A has a symmetric basis, it turns out that it must be commutative. Lemma 3.16. Assume B is symmetric. Then A is commutative. Proof. Let a, b ∈ A. By Lemma 3.15, T
T
T
Mab = Ma Mb = (Ma Mb ) = Mb Ma = Mb Ma = Mba . So, ab = ba. Lemma 3.17. Assume B is symmetric and that σ fixes the elements of B. Let x, y ∈ An where hx, yiAn = 0. Then hΦ(x), Φ(y)iRnr = 0.
12
Proof. Let a = a1 v1 + · · · + ar vr , b = b1 v1 + · · · + br vr ∈ A where ai , bi ∈ R. Since σ fixes the elements of B, (σ(b1 ), . . . , σ(br )) = ρ(σ(b)) = ρ(1)Mσ(b) . So, hρ(a), ρ(b)i
Rr
=
r X
ai σ(bi ) = ρ(1)Ma ρ(1)Mσ(b)
i=1
T
.
Pn Pn Since hx, yi = 0, 0 = i=1 xi σ(yi ) which implies 0 = i=1 Mxi Mσ(yi ) . Since B is symmetric, by Lemma 3.13 and Lemma 3.15 we have hΦ(x), Φ(y)iRnr
n X
=
hρ(xi ), ρ(yi )iRr =
i=1
n X
=
ρ(1)Mxi ρ(1)Mσ(yi )
i=1
T
T
T
ρ(1)Mxi Mσ(yi ) ρ(1)
i=1
=
n X
ρ(1)
n X
Mxi Mσ(yi )
i=1
!
T
ρ(1) = 0
Theorem 3.18. Let C be an n length linear code over A. Assume A and R are Frobenius rings, B is symmetric and σ fixes the elements of B. Then Φ(C ⊥ ) = Φ(C)⊥ . Proof. By Lemma 3.17, since B is symmetric and σ fixes the elements of B, Φ(C ⊥ ) ⊂ Φ(C)⊥ . We know |C| = |Φ(C)|. Since A and R are Frobenius rings, from Lemma 2.3, we have that |C ⊥ | =
|A|n |R|rn = = |Φ(C)⊥ |. |C| |Φ(C)|
In [3] the ring Fq [x]/(xt ) is considered. Among other results, a condition on the change of basis matrix which changes from the standard basis, {1, x, . . . , xt−1 }, to some other basis is given which guarantees that the new basis preserves orthogonality. It turns out that this condition is equivalent to the new basis being symmetric. This is the subject of the next proposition. Proposition 3.19. Assume A = Fq [x]/(xr ) and R = Fq . Let B be a change of basis matrix from the standard Fq -basis for A to the Fq -basis B. Then the following are equivalent: 1. B is symmetric. T
2. BB is upper anti-triangular with constant anti-diagonal as well as all parallel diagonals. T (For bij = BB , if i + j > r + 1 then bij = 0 and if k = i + j ≤ r + 1, ij
b1,k−1 = b2,k−2 · · · = bk−1,1 )
13
Proof. In the following we use the embedding of A in Mr (Fq ) as in Definition 3.12 where for a ∈ A, Ma is its matrix representation Euclidean inner h andT · is the i T T T r product on Fq with respect to B. Then B = ρ(1) , ρ(x) , . . . , ρ(xr−1 ) , T showing BB = ρ(xi−1 ) · ρ(xj−1 ). ij
Assume B is symmetric. Then for m, n ≥ 0 ρ(xm ) · ρ(xn )
T
T
T
= ρ(1)Mxm (ρ(1)Mxn ) = ρ(1)Mxm Mxn ρ(1) T
T
= ρ(1)Mxm Mxn ρ(1) = ρ(1)M1 Mxm+n ρ(1)
T
T
T
= ρ(1)M1 Mxm+n ρ(1) = ρ(1)M1 (ρ(1)Mxm+n ) = ρ(1) · ρ(xm+n ). T
With this it can easily be shown BB is upper anti-triangular with constant anti-diagonals. T Now assume BB is upper triangular with constant anti-diagonals. This condition is equivalent to saying ρ(1)·ρ(xk ) = ρ(x)·ρ(xk−1 ) = · · · = ρ(xk )·ρ(1). In some sense this says that the inner product respects the multiplication in A. Using this condition c ∈ A. be shown that ρ(ab)·ρ(c) = ρ(a)·ρ(bc)for a, b, it can T T we = ρ(vi a) · ρ(vj ) = ρ(vi ) · ρ(vj a) = M1 Ma Let a ∈ A. Since Ma M1 ij
ij
T
T
T
have that Ma = Ma M1 = Ma M1 = M1 Ma = Ma . Hence, B is symmetric.
Now that we see that the condition presented in [3] Proposition 5 is equivalent to a basis being symmetric, it is clear that their condition guarantees that the basis preserves duality which is the essence of Proposition 5 and Corollary 1 in [3]. Example 3.20. Let A = F4 [x]/(x2 ) and B = (1, x + 1) which is an F4 -basis for A. Let B bethe change of basis matrix fromthe standard F4 -basis for A to T 1 0 1 1 B. Then B = and BB = . By Proposition 3.19, this is 1 1 1 0 a symmetric basis. Using the embedding described in Definition 3.12, we have 1 0 0 1 M1 = and Mx+1 = showing directly that B is symmetric. 0 1 1 0 Now, let σ be the Frobenius map on F4 extended linearly to A. Then σ(1) = 1 and σ(1 + x) = 1 + x showing σ fixes the elements of B. By Theorem 3.18, since A is Frobenius, if C is a linear code over A, Φ(C ⊥ ) = Φ(C)⊥ . There are no pseudo-self-dual bases for A over F2 nor F4 . But, A has 18 symmetric bases over F4 = F2 (α) ⊂ A (out of 90 bases over F4 ) (see also [24]): {1, x + 1}, {α, α(x + 1)}, {α2 , α2 (x + 1)} {1, αx + 1}, {α, α2 x + α)}, {α2 , x + α2 )} {1, α2 x + 1}, {α, x + α)}, {α2 , αx + α2 )} {x + 1, αx + 1}, {α(x + 1), α2 x + α}, {α2 (x + 1), x + α2 } 14
{x + 1, α2 x + 1}, {α(x + 1), x + α}, {α2 (x + 1), αx + α2 } {x + α, α2 x + α}, {αx + α2 , x + α2 }, {α2 x + 1, αx + 1} 1. {1, α2 x + 1} is the unique symmetric basis whose elements are invariant under the order 2 automorphism σ1 defined by α 7→ α2 and x 7→ αx. Since σ1 restricts to the Frobenius automorphsim on F4 , the basis {1, α2 x + 1} allows to maps selfdual code for the σ1 -hermitian form over A onto a hermitian self dual code over F4 . 2. {1, αx + 1} is the unique symmetric basis whose elements are invariant under the order 2 automorphism σ2 defined by α 7→ α2 and x 7→ α2 x. Since σ2 restricts to the Frobenius automorphsim on F4 , the basis {1, αx+ 1} allows to maps selfdual code for the σ2 -hermitian form over A onto a hermitian self dual code over F4 . 3. {1, x+1} is the unique symmetric basis whose elements are invariant under the order 2 automorphism σ3 defined by α 7→ α2 and x 7→ x. Since σ3 restricts to the Frobenius automorphsim on F4 , the basis [1, x + 1] allows to maps selfdual code for the σ3 -hermitian form over A onto a hermitian self dual code over F4 . The ring A has also 18 symmetric bases over F2 (out of 840 bases of A over F2 ). One such basis is (1, x + 1, αx + α, α) for which the matrices Mv are 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 , 0 0 0 1 , 1 0 0 1 , 0 1 1 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1
None of these 18 bases have the property that its elements are fixed under an automorphism of order 2 since only the identity can fix the elements of a basis over the prime ring. The ring A has also 3 subrings isomorphic to F2 [x]/(x2 ) (also noted F2 + uF2 in coding theory), one of them is R = F2 [x]/(x2 ) ⊂ F4 [x]/(x2 ). There are 24 symmetric bases of A over R, one such a basis is (1, α) for which 1 0 0 1 M1 = and Mα = 0 1 1 1 Another symmetric basis of A over R is (x + 1, α) for which the matrices Mv are x+1 0 0 x+1 Mx+1 = and Mα = . 0 x+1 x+1 1
Example 3.21. The Galois ring A = GR(4, 2) defined in example 3.7 has 24 symmetric bases (among which are the 8 σ-pseudo-self-dual bases previously found). An example of a new basis is {1, ξ + 2}. Example 3.22. The ring Z4 [x]/(x2 − 2x) (see [26]) has 16 symmetric basis. An example of such a basis is {1, x + 1}. 15
Example 3.23. The ring Z4 [x]/(x2 − 2) has 16 symmetric bases. An example of such a basis is {1, x + 1}. Example 3.24. The ring Z4 [x]/(x2 − x) ([20]) has no σ-pseudo-self-dual bases, but has 8 symmetric bases over Z4 . An example of such a basis is {x + 3, x}. Example 3.25. The ring F2 [x]/(x4 ) has no σ-pseudo-self-dual bases, but has 12 symmetric bases over F2 . An example of such a basis is {x2 +x, x3 +x, 1, x3 +1}. Example 3.26. The ring F2 [x]/(x2 + y 2 , xy) (see [25]) has 16 symmetric bases over F2 . An example of such a basis is y 2 + 1, y, x, 1 .
Example 3.27. The ring A = F2 [x]/(x2 , y 2 ) has 8 symmetric basis over F2 . An example of such a basis is (y + 1, x + 1, xy + x + y + 1, 1). The ring A has also 7 subrings isomorphic to F2 [x]/(x2 ). For the subring R = {0, 1, x, x + 1} there are 48 symmetric bases of A over R. One such a basis is (y + 1, x + 1) for which 0 x+1 x+1 0 My+1 = and Mx+1 = . x+1 0 0 x+1 Example 3.28. The ring F2 [x]/(x3 ) of order 8 hasno self dual basis but has four symmetric bases: x2 + x, x2 + 1, 1 , x2 + 1, x, 1 , x2 + x, x + 1, x2 + x + 1 , x, x + 1, x2 + x + 1
Example 3.29. The ring F2 [x]/(x3 −1) of order 8 has no self dual basis but has three symmetric bases: x2 + 1, x + 1, x2 + x + 1 , x2 + x, x + 1, x2 + x + 1 , x2 + x, x2 + 1, x2 + x + 1 .
Example 3.30. The self dual basis (x, x + 1) of the ring F2 [x]/(x2 + x) of order 4 is also a symmetric basis.
Example 3.31. The ring F2 [x]/(x2 ) of order 4 (cf. ([16])) has no σ-pseudoself-dual basis but has the unique symmetric basis (x + 1, 1). Example 3.32. We give some examples in a characteristic 3. 1. The ring A = F3 [x]/(x2 , y 2 ) who has 2592 symmetric basis over F3 . An example of such a basis is (2x + y + 2, 2xy + 2x + y + 1, xy + x + y, 1). The automorphism group of A is of order 72 and contains 21 elements of order 2, 8 elements of order 3, 18 elements of order 4 and 24 elements of order 6. 2. The ring A = F3 [x]/(x3 ) has 72 symmetric basis over F3 . An example of such a basis is 2x2 + 2x + 1, x + 1, 1 . The automorphism group of A is S3 . 3. The ring F3 [x]/(x2 + x) has 8 symmetric basis over F3 . An example of such a basis is (x + 2, 1). The automorphism group of A is S2 .
16
4. The ring A = F9 [x]/(x2 ), whose automorphism group is D8 , has 576 symmetric basis over F9 . An example of such a basis is 1, x + α6 where α is a generator of F∗9 . has also many symmetric basis over F3 . An example of such a basis is (x + y + 2, xy + x + y + 1, 1, xy + 2x + y). 5. The ring F9 [x]/(x2 + x) has 256 symmetric basis over F9 . An example of such a basis is (x + 1, x). The ring F9 [x]/(x2 +x) has also many symmetric basis over F3 . An example of such a basis is 1, a6 x + a6 , a6 x, x + 2 .
4
Principal skew module codes over rings
Following ([9, 10, 12]) we now introduce the notion of a “principal skew cyclic module” codes C over a ring A which are a generalization of cyclic codes. In [9] principal skew module codes over the Galois ring A = GR(4, 2) where considered (cf. example 3.7). We refer to the above three papers for further proofs and references. Starting from the finite ring A and an automorphism θ of A, we define a ring structure on the set A[X; θ] = {an X n + . . . + a1 X + a0 |ai ∈ A and n ∈ N}. The addition in A[X; θ] is defined to be the usual addition of polynomials and the multiplication is defined by the basic rule X · a = θ(a)X (a ∈ A) and extended to all elements of A[X; θ] by associativity and distributivity. With these two operations A[X; θ] is a ring know as a skew polynomial ring or Ore ring. If the leading coefficient of g ∈ A[X; θ] is invertible, then for any f ∈ A[X; θ] there exists a unique decomposition f = qg + r where either r = 0 or deg(r) < deg(g). Definition 4.1. ([9, 12]) Consider f ∈ A[X; θ] be of degree n. A principal module θ-code C is a left A[X; θ]-submodule A[X; θ]g/A[X; θ]f ⊂ A[X; θ]/A[X; θ]f in the basis 1, X, . . . , X n−1 where g is a monic right divisor of f in A[X; θ]. The length of the code is n = deg(f ) and its dimension is k = deg(f ) − deg(g), we say that the code C is of type [n, k]. If the minimal Hamming distance of the code is d, then we say that the code C is of type [n, k, d]A . We denote this code C = (g)n,θ . If there exists an a ∈ A∗ such that g divides X n − a on the right then the code (g)n,θ is θ-constacyclic. We will denote it (g)an,θ . If a = 1, the code is θ-cyclic and if a = −1, it is θ-negacyclic. For a principal module θ-cyclic code (g)θn of length n over a ring A, we have (c0 , . . . , cn−1 ) ∈ (g)θn ⇒ (θ(cn−1 ), θ(c0 ), . . . , θ(cn−2 )) ∈ (g)θn . When θ is the identity we obtain the classical cyclic codes, showing that principal module θ-cyclic code are a natural generalization of cyclic codes. Note that a submodule A[X; θ]g/A[X; θ]f ⊂ A[X; θ]/A[X; θ]f where g is not monic will in general not be a free A[X; θ]-modules. Pn−k For g = i=0 gi X i , the generator matrix of a module θ-code (g)n,θ is given 17
by Gg,n,θ = g0 ... gn−k−1 0 θ(g0 ) ... . .. . 0 . . 0 0 ... 0
gn−k θ(gn−k−1 ) .. .
0 θ(gn−k ) .. .
θk−1 (g0 )
...
... ... .. .
0 0 .. .
θk−1 (gn−k−1 ) θk−1 (gn−k )
In many cases, A[X; θ] is not a unique factorization domain, therefore there can be many divisors of X n − 1 leading to many principal skew cyclic module codes as well as many classical cyclic codes. In the following table we give, depending on the ring A, the number of distinct monic right divisors of degree r of X n − 1 for n = 2r in A[X; θ]. The case θ = id corresponds to A[X] and therefore to classical cyclic codes over A, while the case θ 6= id is the total number of all right divisors of degree n/2 in all rings A[X; θ] where θ 6= id
n 2 4 6 8
F2 [x,y] A = (x 2 ,y 2 ) |A| = 16 Aut(A) ∼ = S4 θ = id 6= id 8 64 64 608 512 1648 4096 30848
A = F(x4 [x] 2) 16 Aut(A) ∼ = S3 = id 6= id 4 20 16 122 88 680 256 3074
2 [x,y] A = (x2F+y 2 ,xy) |A| = 16 Aut(A) ∼ = D4 = id 6= id 4 28 32 256 64 528 1024 9216
A = F(x2 [x] 4) |A| = 16 Aut(A) ∼ = D2 = id 6= id 4 16 16 80 64 384 512 1536
A = M (2, F2 ) |A| = 16 Aut(A) ∼ = S3 = id 6= id 4 14 16 50 76 380 256 770
Table 1: Number of monic factors of degree n/2 of X n − 1 ∈ A[X; θ] Example 4.2. Let A = M2 (F2 ). The 66 divisors of degree 2 of f = X 4 − 1 ∈ A[X; θ] in table 4 lead to 40 distinct generator matrices and therefore to 40 different codes [4, 2]A . a b d c Taking for θ the automorphism 7→ of order 2, the skew c d b a polynomial 1 0 0 1 1 1 2 g= X + X+ ∈ A[X; θ] 0 1 1 1 1 0 is one of the 16 second order right factors of f ∈ A[X; θ]. The decomposition in A[X; θ] is 0 1 0 1 f = X2 + X+ ·g 1 1 1 1 The generating matrix G of the [4, 2]A principal skew module code A[X; θ]g/A[X; θ]f ⊂
18
A[X; θ]/A[X; θ]f is 1 1 0 0
1 0 0 0
0 1 0 1
1 1 0 0 0 1 0 1 0 0 . 1 1 1 1 0 1 1 0 0 1
In order to obtain the generating matrix over A[X; θ] of the image code Φ(C), where Φ is defined as in section 3, we note that the code C corresponding to the A module A[X; θ]g/A[X; θ]f ⊂ A[X; θ]/A[X; θ]f is spanned by A-multiples Pof the lines ofthe above matrix G, which are of the form n−k j i+j a·(X j g) = a , where a ∈ A. For an R-basis B = {v1 , . . . , vr } i=0 θ (gi )X P r α v · (X j g), and therefore spanned over of A we get that C is spanned i i j=1 Pn−k R by the lines corresponding to i=0 (vi θj (gi ))X i . Therefore the lines of the generating matrix of Φ(C) are given by the image under Φ of the vectors corresponding to the polynomials v1 g, · · · , vn g, v1 Xg, · · · , vn Xg, · · · , v1 X n−deg(g) g, · · · , vn X n−deg(g) g Example 4.3. Using the standard basis 1 0 0 1 0 0 0 0 , , , 0 0 0 0 1 0 0 1 of A = M2 (F2 ) over F2 , the [4, 2]A code C from the previous example 4.2 is mapped under Φ to a [16, 8, 4]F2 code Φ(C), where Φ is defined as in section 3, whose generator matrix is 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 The first line corresponds to the image under Φ of the coefficients of 1 0 1 1 0 1 1 0 ·g = + X+ X2 0 0 0 0 0 0 0 0 Instead of R = F2 we now consider the subfield 0 0 1 0 1 1 0 F4 ∼ ,1 = ,α = , α2 = =R= 0= 0 0 0 1 1 0 1 19
(4.1)
1 1
1 0 0 0 , v2 = , then 0 0 0 1 the [4, 2]A code C from the previous example 4.2 is mapped under Φ, where Φ is defined as in section 3, to a [8, 4, 4]F4 code Φ(C) whose generator matrix is 1 α 0 α 1 0 0 0 α2 0 α2 1 0 1 0 0 0 0 0 α 1 α 1 0 0 0 α2 1 α2 0 0 1 and the R-basis of M2 (F2 ) given by
v1 =
Againthe firstrow corresponds tothe images of (4.1) under ρ of the coefficients 1 1 0 1 1 0 since = 1 · v1 + αv2 , = 0 · v1 + αv2 and = 0 0 0 0 0 0 1 · v1 + 0 · v2 . Rings with many automorphisms produce more principal skew module codes.
Example 4.4. Consider F4 = F2 (α), A = F4 [x]/(x2 ) defined in example 3.20 and θ ∈ Aut(A) ∼ = S3 of order 2 defined by θ(α) = α2 and θ(x) = x. The θ-skew cyclic codes of type [2, 1]A are generated by one of the 6 monic right divisors X + α2 , X + (x + 1), X + 1, X + (αx + α), X + (α2 x + α2 ), X + α of X 2 − 1 ∈ A[X; θ]. If we use the symmetric F4 -basis α2 x + 1, αx + 1 of A, then the best [4, 2]4 code obtained under the mapping Φ is of hamming distance 2. However, if we use the the symmetric F4 -basis 1, α2 x + 1 of A, then the best [4, 2]4 code obtained under the mapping Φ is of hamming distance 3. For instance in the basis F4 -basis 1, α2 x + 1 the mapping of the [2, 1]A code generated by X + (x + 1) ∈ A[X; θ] is generated by 2 α α 1 0 α α2 0 1 and is of hamming distance 3. Therefore the set of image codes obtained under mapping (and the best possible hamming distance) depends of the chosen F4 basis.
5
Self dual codes
Using a basis for a Frobenius ring A over a subring R that preserves duality as described in Section 3, the image code Φ(C) of a self-dual code C over A is guaranteed to be self dual code over R. Having a method for constructing selfdual codes over A then, we may find self-dual codes over R. In this section, the setting will be the following. We start with a finite Frobenius commutative ring A which is an R-algebra with a duality preserving R-basis where R a subring of A. We focus on skew cyclic codes over A and use Corollary 6 of [13] which provides a method for finding Euclidean self-dual skew module codes over a 20
commutative ring A. For examples of classical constructions for self-dual cyclic and θ-cyclic codes which are well-known examples of skew-module codes, see [1, 7, 22, 23, 30]. Algebraic characterizations of self-dual cyclic codes exists also for noncommutative rings in [15]. Definition 5.1 (cf. [13]). Let A be a commutative ring and Pm θ be an automorphism on A. The skew reciprocal polynomial of h = i=0 hi X i ∈ A[X; θ] of degree m is m m X X θi (hm−i ) X i . X m−i · hi = h∗ = i=0
i=0
The left monic skew reciprocal polynomial of h is h♮ := (1/θm (h0 )) · h∗ .
When θ is the identity we obtain again the classical reciprocal polynomial. Since θ is an automorphism, the map ∗ : A[X; θ] → A[X; θ] given by h 7→ h∗ is a bijection. In particular for any g ∈ A[X; θ] there exists a unique h ∈ A[X; θ] such that g = h∗ and, if g is monic, such that g = h♮ . Corollary 5.2 (cf. [13]). Let A be a commutative ring and θ be an automorphism on A. A module θ-code (g)θ2k with g ∈ A[X; θ] of degree k is self-dual if and only if there exists h ∈ A[X; θ] such that g = h♮ and h♮ h = X 2k − ε with ε ∈ {−1, 1}. According to the table given in [11], the self dual skew module codes [16, 8]F4 over F4 constructed directly in F4 [X; θ] (i.e. starting with a ring of order 4) are of minimal distance at most 4. The next example shows that using rings of order 16 in order to obtain codes over F4 produces more codes, among which is an optimal [16, 8, 6]F4 self dual code (cf. [19]). Example 5.3. Consider F4 = F2 (α), A = F4 [x]/(x2 ) defined in example 3.20. For each θ ∈ Aut(A) ∼ = S3 we consider the self-dual θ-cyclic codes of length 6 over A which can be characterized using the previous corollary. The number ♯θ of polynomial that generate a self-dual θ-cyclic code over A of length 6 depends on the order of θ |θ| 1 3 2 ♯θ 24 36 12 For θ ∈ Aut(A) of order 1 (classic cyclic codes) and 3 the best hamming distance obtained under the mapping Φ using all the symmetric F4 -bases given in example 3.20 is 4. For θ ∈ Aut(A) of order 2 given by α 7→ α2 ; x 7→ αx the self-dual θ-cyclic code over A generated by the right divisor X 3 + (α2 x + α)X 2 + (αx + α)X + (α2 x + 1) of X 6 − 1 ∈ A[X; θ] is mapped under Φ and the F4 -basis α2 x + 1, αx + 1 to
21
the [12, 6, 6]4 self α α2 0 0 0 0
dual code over F4 generated by
0 0 0 0 0 0 0 0 0 1 0 0 α2 1 0 1 0 1 However, if we use the the symmetric F4 -basis 1, α2 x + 1 of A, then the best self dual code over F4 code obtained under the mapping Φ is of hamming distance 4, illustrating again the set of image codes obtained under mapping (and the best possible hamming distance) depends of the chosen F4 -basis. Using again the F4 -basis α2 x + 1, αx + 1 of A and θ of order 2 given by α 7→ α2 ; x 7→ αx we obtain for the right divisor g = X 4 + (α2 x + α2 )X 3 + α2 X 2 + α2 X + α2 x + α of X 8 − 1 in A[X; θ] a [16, 8, 6]F4 self dual linear code with generator matrix 1 α2 α2 0 α2 0 0 α2 1 0 0 0 0 0 0 0 α2 1 0 α2 0 α2 α2 0 0 1 0 0 0 0 0 0 2 2 0 0 0 α α 0 α 0 1 α 1 0 0 0 0 0 0 0 α2 0 0 α 0 α α2 1 0 1 0 0 0 0 0 0 0 0 1 α2 α2 0 α2 0 0 α2 1 0 0 0 0 0 0 0 α2 1 0 α2 0 α2 α2 0 0 1 0 0 0 0 0 0 0 0 0 α2 α 0 α 0 1 α2 1 0 0 0 0 0 0 0 α2 0 0 α 0 α α2 1 0 1 α2 α 0 0 0 0
0 α α α2 0 0
α 0 α2 α 0 0
1 α2 α 1 α α2
α2 1 1 α α2 α
1 0 0 α2 0 α
0 1 α2 0 α 0
0 0 1 0 1 α2
Table 2 gives, depending on the length n = 2k of the code, the number of skew polynomials of degree k who satisfy the criteria of corollary 5.2 in some ring A[X; θ] and generate a θ-cyclic self-dual code over A. Here we count the polynomials, not the different codes obtained. The number of such polynomials is connected to the size of the ring A and to the size of the automorphism group of A since each automorphism gives a new skew polynomial ring with a different arithmetic. The identity automorphism corresponds to the classical commutative polynomial ring over A and the codes are cyclic codes in this case. The number of polynomials also depends on other properties of the ring like the number of monic factors of X n − 1 ∈ A[X; θ] (cf. Table 2). Table 3 compares the best hamming distance d of the self-dual binary codes of type I and II obtained under the mapping Φ to the hamming distance d in the tables (cf. [19]. The index θ indicates that this hamming distance for the mapping can only be obtained using skew polynomials A[X; θ] where θ 6= id, i.e. this distance cannot be obtained using classical cyclic codes in A[X]. The best distance of a binary self-dual code of length 24 of type I and II obtained by mapping a classical cyclic (where the automorphism is the identity) self-dual code over F2 [x, y]/(x2 , y 2 ), F4 [x]/(x2 ) is 4. The best distance of a binary self-dual code of length 32 of type I and II obtained by mapping a classical cyclic self-dual code over F4 [x]/(x2 ) is 4. 22
length over A 2 4 6 8 10 12
F2 [x,y] A = (x 2 ,y 2 ) |A| = 16 Aut(A) ∼ = S4 θ = id 6= id 8 34 40 344 60 488 320 3328 512 668 2560 20480
length over A 4 8 12 16 20 24
A = F(x4 [x] 2) |A| = 16 Aut(A) ∼ = S3 = id 6= id 4 8 16 50 24 108 64 290 64 122 416 1920
2 [x,y] A = (x2F+y 2 ,xy) |A| = 16 Aut(A) ∼ = D4 = id 6= id 4 20 24 104 16 88 128 1152 64 336 768 3584
A = F(x2 [x] 2) |A| = 4 Aut(A) ∼ = {id} θ = id 6= id 4 − 8 − 16 − 32 − 64 − 128 −
A = F(x2 [x] 4) |A| = 16 Aut(A) ∼ = D2 = id 6= id 4 12 16 64 32 64 96 288 64 256 256 1792
[x] A = (xF22+x) |A| = 4 Aut(A) ∼ = S2 = id 6= id 1 1 1 3 1 3 1 11 1 9 1 53
Table 2: Number of factors of X n − 1 ∈ A[X; θ] that generate a self-dual code
length over F2 8 16 24 32 40 48
Gaborit table I II 2 4 4 4 6 8 8 8 8 8 10 12
F2 [x,y] A = (x 2 ,y 2 ) |A| = 16 I II 2 4 4 4 6θ 8θ 8 8 8 8 8 8
F4 [x] (x2 )
F2 [x,y] (x2 +y 2 ,xy)
16 I 2 4 4 8θ 8θ 8
II 4 4 8θ 4 8θ 8
I 2 4 4 4 4 4
16 II 4 4 4 4 4 4
F2 [x] (x4 )
F2 [x] (x2 )
16
4 I II 2 4 4 4 4 4 4 4 4 4 4 4
I 2 4 4 8 8θ 8
II 4 4 4 4 8θ 8
F2 [x] (x2 +x)
4 I 2 2 4θ 4θ 4θ 4θ
II − 4θ − 4θ − 8θ
Table 3: Best hamming distance for self-dual binary code under mapping. Example 5.4. Self-dual skew cyclic codes over A = GR(4, 2) are classified in [9]. There are 8 generators of self-dual codes of length 4 that divide X 4 − 1 ∈ GR(4, 2)[X; θ] (for θ 6= id, there are none in GR(4, 2)[X]) among them X 2 + (w + 1)X + 3w which is of minimal Lee distance 6 and therefore optimal 23
(cf. [22]). There are no generators of self-dual skew cyclic codes of length 8 over GR(4, 2) (whose generator divides X 8 − 1 ∈ GR(4, 2)[X; θ]). There are 192 generators of self-dual skew cyclic codes of length 8 over GR(4, 2) (whose generator divides X 10 − 1 ∈ GR(4, 2)[X; θ]) and the best Lee distance obtained is 10 while the optimal Lee distance is 12 (cf. [22]) Example 5.5. Consider A = Z4 [x]/(x2 − 2). The automorphism group is isomorphic to D2 of order 4. There are 8 generators of self-dual codes of length 4 that divide X 4 − 1 ∈ A[X; θ] for θ = id and for θ defined by x 7→ 3x. An example is X 2 +xX +2x+3 ∈ A[X], which, in the symmetric Z4 -basis (3x+1, 1) of A is mapped to a Z4 -code of length 8 of optimal Lee distance 6 (cf. [22]) whose generating matrix is the same as in the previous example. for each θ ∈ Aut(A) there are 32 generators self-dual cyclic codes of length 8 that divide X 8 − 1 ∈ A[X] for θ = id. An example is X 4 + (2x + 2)X 3 + (3x + 2)X 2 + (2x + 2)X + 2x + 3 ∈ A[X], which, in the symmetric Z4 -basis (3x + 1, 1) of A is mapped to a Z4 -code of length 16 of optimal minimal Lee distance 8 (cf. [22]) whose generating matrix is 1 2 0 2 3 1 0 2 1 0 0 0 0 0 0 0 2 1 2 0 1 1 2 0 0 1 0 0 0 0 0 0 0 0 1 2 0 2 3 1 0 2 1 0 0 0 0 0 0 0 2 1 2 0 1 1 2 0 0 1 0 0 0 0 0 0 0 0 1 2 0 2 3 1 0 2 1 0 0 0 0 0 0 0 2 1 2 0 1 1 2 0 0 1 0 0 0 0 0 0 0 0 1 2 0 2 3 1 0 2 1 0 0 0 0 0 0 0 2 1 2 0 1 1 2 0 0 1
Its Lee-weight distribution is 1 + 508z 8 + 896z 10 + 10752z 12 + . . .. Another example is X 4 + 2X 3 + xX 2 + 2X + 3, ∈ A[X], which, in the symmetric Z4 -basis (3x + 1, 1) of A is mapped to a Z4 -code of length 16 of optimal minimal Lee distance 8 whose generating matrix is 3 0 2 0 3 3 2 0 1 0 0 0 0 0 0 0 0 3 0 2 3 1 0 2 0 1 0 0 0 0 0 0 0 0 3 0 2 0 3 3 2 0 1 0 0 0 0 0 0 0 0 3 0 2 3 1 0 2 0 1 0 0 0 0 0 0 0 0 3 0 2 0 3 3 2 0 1 0 0 0 0 0 0 0 0 3 0 2 3 1 0 2 0 1 0 0 0 0 0 0 0 0 3 0 2 0 3 3 2 0 1 0 0 0 0 0 0 0 0 3 0 2 3 1 0 2 0 1
Its Lee-weight distribution is 1 + 380z 8 + 1920z 10 + 7168z 12 + . . .
Example 5.6. For F25 = F5 (α) where α2 + 4α + 2 = 0 the polynomial X 4 + α9 X 3 + α2 X 2 + αX + α16 ∈ F25 [X; θ] (where θ : a 7→ a5 is the Frobenius automorphisms) is a right factor of X 8 + 1 ∈ F25 [X; θ] and generates a self-dual code C over F25 . Using the symmetric F5 -basis (α5 , α7 ) the code C is mapped 24
to an optimal self-dual code Φ(C) over F5 and whose generating matrix is 4 3 2 1 0 1 2 3 1 3 0 1 4 1 2 3 3 0 0 0 0 2 4 4 2 4 3 0 0 2 4 4 2 4 0 2 0 0 0 0 4 3 2 1 0 0 0 0 0 3 0 1 4 1 0 0 0 0 0 0 0 2 4 0 0 0 0 0 0 2 4 4
(cf. [19]) with minimal distance 7 0 1 2 2 1 2 4 2
0 0 1 0 2 3 2 4
0 0 0 1 3 3 4 0
0 0 0 0 1 0 3 2
0 0 0 0 0 1 2 2
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
and weight distribution 1 + 448z 7 + 3360z 8 + 4992z 9 + . . . The best best image of a self-dual negacyclic code (whose generator divides X 8 + 1 in the classical commutative polynomial ring) is of minimal distance 4. The polynomial X 5 + α11 X 4 + α22 X 3 + α16 X 2 + α17 X + 3 ∈ F25 [X; θ] generates a self-dual negacyclic skew module code C over F25 . Using the above symmetric F5 -basis the code C is mapped to an optimal self-dual code Φ(C) over F5 (cf. [19]) with minimal distance 8 and weight distribution 1 + 1280z 8 + 3200z 9 + 24848z 10 + . . .. There are no other possible weight distribution for a self-dual code Φ(C) over F5 with minimal distance 8 which is the image of a self-dual negacyclic skew code. The best image of length 20 of a self-dual negacyclic code of length 10 over A (using the classical commutative polynomial ring) is of minimal distance 4. Example 5.7. The automorphism group of the ring A = F5 [x]/(x2 ) of order 25 is isomorphic to the cyclic group C4 of order 4. This leads to 4 skew polynomial rings, one of which is the standard commutative polynomial ring corresponding to the identity. For γ : x 7→ 4x of order 2 the polynomial X 5 + (4x + 4)X 4 + 3X 3 + X 2 + (2x + 2)X + 2 ∈ A[X; γ] generates a self-dual negacyclic skew module code C of length 10 over A = F5 [x]/(x2 ). Using the symmetric F5 basis (x + 2, 1) the code C is mapped to an optimal self-dual code Φ(C) of length 20 over F5 (cf. [19]) with minimal distance 8 and weight distribution 1 + 1380z 8 + 2880z 9 + 24704z 10 + . . .. The best image of length 20 of a self-dual negacyclic code of length 10 over A (using the classical commutative polynomial ring) is of minimal distance 7.
Acknowledgements The first author would like to thank IRMAR for the support of his visit to their institution during which time this work was initiated.
References [1] Taher Abualrub, Nuh Aydin, and Padmapani Seneviratne, On θ-cyclic codes over F2 +vF2 , Australas. J. Combin. 54 (2012), 115–126. MR 3013244 25
[2] Adel Alahmadi, Houda Sboui, Patrick Sol´e, and Olfa Yemen, Cyclic codes over M2 (F2 ), J. Franklin Inst. 350 (2013), no. 9, 2837–2847. MR 3146951 [3] R. Alfaro and K. Dhul-Qarnayn, Constructing self-dual codes over Fq [u]/(ut ), Des. Codes Cryptogr. 74 (2015), no. 2, 453–465. MR 3302667 [4] Christine Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory Ser. A 78 (1997), no. 1, 92–119. MR 1439633 (98a:11084) [5] Katherine G. Bartley and Judy L. Walker, Algebraic geometric codes over rings, Advances in algebraic geometry codes, Ser. Coding Theory Cryptol., vol. 5, World Sci. Publ., Hackensack, NJ, 2008, pp. 323–361. MR 2509128 (2010j:14051) [6] Koichi Betsumiya, San Ling, and Fidel R. Nemenzo, Type II codes over F2m + uF2m , Discrete Math. 275 (2004), no. 1-3, 43–65. MR 2026275 (2004i:94051) [7] Maheshanand Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr. 62 (2012), no. 1, 85–101. MR 2873109 (2012k:94202) [8] A. Bonnecaze, E. Rains, and P. Sol´e, 3-colored 5-designs and Z4 -codes, J. Statist. Plann. Inference 86 (2000), no. 2, 349–368, Special issue in honor of Professor Ralph Stanton. MR 1768278 (2001g:05021) [9] Delphine Boucher, Patrick Sol´e, and Felix Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun. 2 (2008), no. 3, 273–292. MR MR2429458 [10] Delphine Boucher and Felix Ulmer, Codes as modules over skew polynomial rings, Cryptography and coding, Lecture Notes in Comput. Sci., vol. 5921, Springer, Berlin, 2009, pp. 38–55. MR 2775615 (2012f:94161) [11]
, Coding with skew polynomial rings, J. Symbolic Comput. 44 (2009), no. 12, 1644–1656. MR 2553570 (2010j:94085)
[12]
, A note on the dual codes of module skew codes, Cryptography and coding, Lecture Notes in Comput. Sci., vol. 7089, Springer, Heidelberg, 2011, pp. 230–243. MR 2945261
[13]
, Self-dual skew codes and factorization of skew polynomials, J. Symbolic Comput. 60 (2014), 47–61. MR 3131378
[14] A. R. Calderbank, A. Roger Hammons, Jr., P. Vijay Kumar, N. J. A. Sloane, and Patrick Sol´e, A linear construction for certain Kerdock and Preparata codes, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 218–222. MR MR1215307 (94b:94020)
26
[15] Pierre-Louis Cayrel, Christophe Chabot, and Abdelkader Necer, Quasicyclic codes as codes over rings of matrices, Finite Fields Appl. 16 (2010), no. 2, 100–115. MR 2594508 (2011e:94208) [16] Steven T. Dougherty, Philippe Gaborit, Masaaki Harada, and Patrick Sol´e, Type II codes over F2 + uF2 , IEEE Trans. Inform. Theory 45 (1999), no. 1, 32–45. MR 1677846 (2000h:94053) [17] Philippe Gaborit, Ann Marie Natividad, and Patrick Sol´e, Eisenstein lattices, Galois rings and quaternary codes, Int. J. Number Theory 2 (2006), no. 2, 289–303. MR 2240231 (2007i:11093) [18]
, Eisenstein lattices, Galois rings and quaternary codes, Int. J. Number Theory 2 (2006), no. 2, 289–303. MR 2240231 (2007i:11093)
[19] Philippe Gaborit and Ayoub Otmani, Tables of self-dual codes, http://www.unilim.fr/pages_perso/philippe.gaborit/SD/index.html. [20] Jian Gao and Yun Gao, Some results on linear codes over Z4 + Z4 v, arXiv: http://arxiv.org/abs/1402.6771 (2014). [21] Marcus Greferath, Cyclic codes over finite rings, Discrete Math. 177 (1997), no. 1-3, 273–277. MR 1483452 [22] W. Cary Huffman, On the classification and enumeration of self-dual codes, Finite Fields Appl. 11 (2005), no. 3, 451–490. MR 2158773 (2006h:94253) [23] Somphong Jitman, San Ling, and Patanee Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun. 6 (2012), no. 1, 39–63. MR 2885847 (2012k:94203) [24] San Ling and Patrick Sol´e, Type II codes over F4 + uF4 , European J. Combin. 22 (2001), no. 7, 983–997. MR 1857260 (2002k:94040) [25] Edgar Mart´ınez-Moro and Steve Szabo, On codes over local Frobenius nonchain rings of order 16, Noncommutative Rings and Their Applications (Steven Dougherty, Alberto Facchni, Andr´e Leroy, Edmund Puczylowski, and Patrick Sol´e, eds.), Contemp. Math., vol. 634, Amer. Math. Soc., Providence, RI, 2015, pp. 227–243. [26] Edgar Mart´ınez-Moro, Steve Szabo, and Bahattin Yildiz, Linear codes over Z4 [x]/hx2 + 2xi, to appear in Int. J. Inform. Coding Theory (2015). [27] C. Mouaha, On q-ary images of self-dual codes, Appl. Algebra Engrg. Comm. Comput. 3 (1992), no. 4, 311–319. MR 1325763 (96e:94019) [28] Gadiel Seroussi and Abraham Lempel, Factorization of symmetric matrices and trace-orthogonal bases in finite fields, SIAM J. Comput. 9 (1980), no. 4, 758–767. MR 592766 (81k:15019)
27
[29] Jay A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555–575. MR MR1738408 (2001d:94033) [30] Bahattin Yildiz and Suat Karadeniz, Linear codes over F2 + uF2 + vF2 + uvF2 , Des. Codes Cryptogr. 54 (2010), no. 1, 61–81. MR 2576881 (2011a:94126)
28
