Minimal codes in binary abelian group algebras - IEEE Xplore

2011 IEEE Information Theory Workshop

Minimal codes in binary abelian group algebras Raul Antonio Ferraz and C´esar Polcino Milies

Marinˆes Guerreiro Departamento de Matem´atica Universidade Federal de Vic¸osa 36570-000 - Vic¸osa - MG, Brasil Email: [email protected]

Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo S˜ao Paulo -SP, Brasil Email: [email protected], [email protected]

Abstract—We give counterexamples to show that some results regarding equivalence of abelian group codes, that have been in the literature for quite some time, are not correct. Also, we give examples of special families of abelian groups for which these results do hold.

I. I NTRODUCTION It is well-known that cyclic codes of length n over a finite field F can be presented both as ideals in the ring Rn = F[X]/hX n − 1i or as ideals in the group algebra FCn , where Cn denotes the cyclic group of order n. Berman [1] and MacWilliams [5] extended this notion to define an abelian code as a proper ideal in the group Fq G, where Fq denotes the field with q elements and G is a finite abelian group. Following Miller [6] we say that two codes C1 and C2 are equivalent if there exists an automorphism θ of G whose linear extension to F2 G maps C1 onto C2 . That same paper includes the following results: Theorem A [6, Theorem 3.6] Let G be a finite abelian group of odd order and exponent n and denote by τ (n) the number of divisors of n. Then there exist precisely τ (n) inequivalent minimal codes in F2 G. Theorem B [6, Theorem 3.9] Let G be a finite abelian group of odd order. Then two minimal abelian codes in F2 G are equivalent if and only if they have the same weight distribution. Unfortunately these statements are not correct. The errors arise from the assumption, implicit in the last paragraph of [6, p. 167], that if G = F2 [Cm ×Cn ] and e and f are primitive idempotents of F2 Cm and F2 Cn , respectively, then ef is a primitive idempotent of F2 G. The actual situation is much more complicated as pointed by Blake and Mullin [2, p. 239]: “A necesssary and sufficient condition to determine those that are primitive, however, seems to be unavailable at the moment.” As we shall see, the primitive idempotents in one such direct product depend on the structure of the lattice of the subgroups of the given group G and on some number theoretical facts on the order of G. We recall the following. Let A be an abelian p-group and F a finite field whose characteristic does not divide the order of A. For

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a subgroup H of A, denote by |H| the number of elb = 1 P ements of H, H h∈H h and, for an element |H| c x ∈ A, set x b = hxi. For each subgroup H of A such that

A/H 6= {1} is cyclic, we can construct an idempotent of FA. In fact, we remark that, since A/H is a cyclic p-group, there exists a unique subgroup H ∗ of A containing H such that b −H c∗ . |H ∗ /H| = p. Define the idempotent eH = H We denote by o(x) the order of an element x in a group. The next result follows immediately from [3, Theorem 4.1]. Theorem 1.1: Let p be an odd prime and A be an abelian p-group of exponent pn . Then the set of idempotents b ∪ {eH = H b −H c∗ | H ≤ A and A/H 6= {1} is cyclic} {A} is the set of primitive idempotents of F2 A if and only if o(¯2) = Φ(pn ) in U (Zpn ).

In the next section we shall list some preliminary results about idempotents that will be needed in the sequel. In Section III, we study minimal abelian codes arising from groups of the type Cpn × Cp , for p a prime number and n a positive integer, and we exhibit counterexamples for Theorems A and B. In Section IV we study minimal abelian codes arising from groups of the type Cpn ×Cpn , for p a prime number and n a positive integer. In this situation Theorem A holds. For these cases, we describe all minimal abelian codes, their parameters and equivalence classes. We shall say that two subgroups H and K of a group G are G-isomorphic if there exists an automorphism ϕ ∈ Aut(G) such that ϕ(H) = K. We establish a correspondence between the equivalence classes of minimal codes and the classes of G-isomorphic subgroups of these groups, showing how the structure of the lattice of the subgroups determines the minimal codes. II. S UBGROUPS , I DEMPOTENTS AND AUTOMORPHISMS The proofs of the results in this section are rather long and mostly group-theoretical; they will be published elsewhere. We shall say that a subgroup H of an abelian group G gives rise to a primitive idempotent in F2 G if there exists a b −H c∗ is a subgroup H ∗ as in Theorem 1.1 such that eH = H primitive idempotent. A characterization of subgroups that give rise to primitive idempotents which will be repeatedly used throughout the paper is given in the sequel.

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Lemma 2.1: Let F be a finite field, K a subgroup of a finite abelian group G such that char(F) does not divide |G|, H1 and H2 subgroups of G containing K and not containing b −H ci , for i = 1, 2, are one another. Then the elements K idempotents which are not primitive in FG. Lemma 2.2: Let A be a finite abelian p-group and H ≤ A. Then A/H is a cyclic group if and only if there exists a unique subgroup L such that H < L ≤ A and [L : H] = p. According to Lemmas 2.1 and 2.2, the lattice of the subgroups of a finite abelian group helps us to identify the subgroups which give rise to primitive idempotent. However, different subgroups may give rise to primitive idempotents which generate equivalent codes. Notice that if H is a subgroup of G which gives rise to a b H c∗ and K is another subgroup primitive idempotent eH = H− of G such that ϕ(H) = K, for an automorphism ϕ of G, b − ϕ(H c∗ ) = K b −K c∗ is also a primitive then ϕ(eH ) = ϕ(H) idempotent. (We use the same notation for ϕ ∈ Aut(G) and its linear extension to F2 G.) Hence, in order to study the equivalence of codes, we need to understand how the group of automorphisms Aut(G) acts on the lattice of the subgroups of G. We observe that isomorphic subgroups are not necessarily G-isomorphic. For example, for a prime p, if G = hai × hbi with o(a) = p2 and o(b) = p, then hap i and hbi are isomorphic but not G-isomorphic, since hbi is contained properly only in hap i × hbi and hap i is contained in hai and in hai bi, for 1 ≤ i ≤ p − 1. However, we have the following. Lemma 2.3: [4, Theorem 3.6] Let n be a positive integer. The automorphism group of the group G = Cn × Cn is isomorphic to GL(2, Zn ). Moreover, if G = hai × hbi, with a 7−→ ai bj o(a) = o(b) = n and θ ∈ Aut(G), then θ : b 7−→ as bt   i j , n = 1. where gcd s t Using this lemma the following can be proved. Lemma 2.4: Let p be a prime integer and r a positive integer. Let G = hxi × hyi, with o(x) = o(y) = pr . Then isomorphic subgroups of G are G-isomorphic.

III. C OUNTEREXAMPLES Proposition 3.1: Let p be an odd prime such that ¯2 generates U (Zp2 ) and G = hai × hbi an abelian group, with o(a) = p2 and o(b) = p. Then F2 G has four inequivalent minimal codes, namely, the ones generated by the idempob e1 = bb − hap\ b and tents e0 = G, i × hbi, e2 = b a−G p \ b e3 = ha i × hbi − G. Also all minimal codes (ideals) of F2 G are described in the

following table with their dimension and mininum weight. Code I0 I1 I1j I2 I2i I3

Primitive Idempotent b e0 = b ab b=G p \ b e1 = b − ha i × hbi jp b − hap \ e1j = ad i × hbi j = 1, . . . , p − 1 b e2 = b a−G c i b e2i = ab − G i = 1, . . . , p − 1 b e3 = hap\ i × hbi − G

Dimension

Mininum Weight

1 2 p −p p2 − p

p3 2p 2p

p−1 p−1

2p2 2p2

p−1

2p2

Proof: In order to use Theorem 1.1, first we need to find all subgroups H of G such that G/H is cyclic. Notice

 that the p + 1 distinct subgroups of order p2 of G are abi , for i = 0, . . . , p − 1, and hap i ×  hbi. The p + 1 distinct subgroups of order p of G are  ajp b , for j = 0, . . . , p − 1, and hap i. The subgroups ajp b , for all j = 0, . . . , p − 1, are contained only in hap i × hbi and hap i is contained in all subgroups of order p2 . Besides, all quotients of G by these subgroups are cyclic, except G/ hap i which is the unique noncyclic quotient of G. The quotient of G by hap i × hbi is also cyclic. Now applying Theorem 1.1, we have the following minimal codes generated by primitive idempotents. b and dim I0 = 1. The code I0 = F2 G · e0 , where e0 = G As hbi is uniquely contained in hap i × hbi, we have I1 = F2 G · e1 , where e1 = bb − hap\ i × hbi, and dim I1 = φ(p2 ) = p2 − p. The codes I1j = F2 G · e1j , where jp b − hap \ e1j = ad i × hbi, for all j = 1, . . . , p − 1, are all equivalent to I1 , since the extension to the group algebra F2 G of the isomorphism ψj : G → G given by ψ(a) = a and ψ(b) = ajp b, for each j, maps I1 onto I1j . b and I3 = F2 G · e3 , Let I2 = F2 G · e2 , where e2 = b a − G, b We have dim I3 = dim I2 = where e3 = hap\ i × hbi − G. φ(p) = p − 1. We also have the codes I2i = F2 G · e2i , where ci − G, b for i = 1, . . . , p − 1, all equivalent to I2 with e2i = ab corresponding isomorphism ϕi : G → G given by ϕ(a) = abi and ϕ(b) = b. We prove now that the codes Ik , with k = 0, 1, 2, 3, are four inequivalent minimal codes in F2 G. It is obvious that I0 is not equivalent to any of the other codes Ik , for k 6= 0, and also that I1 is not equivalent to either I2 or I3 . Let us prove that I2 and I3 are inequivalent. Notice that supp(e2 ) = G \ hai, which contains elements of order p, and supp(e3 ) = G \ hap i × hbi, which only contains elements of order p2 . Hence, if there is an isomorphism ψ : G → G such that ψ(e2 ) = e3 , we would have elements of order p being mapped to elements of order p2 , a contradiction. Therefore, I2 is not equivalent to I3 . For the mininum weight of the codes, it is clear that the minimal code I0 has minimum weight p3 , as all its nonzero elements have this weight. For 1 ≤ j 6= k ≤ p−1, as supp(ajpbb)∩supp(akpbb) = ∅, the element (ajp + akp )e1 = (ajp + akp )bb is in I1 and has weight 2p. Notice that I1 ⊂ F2 G · bb, thus the weight of any element

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of I1 must be a positive multiple of p. Hence, if there is an element in I1 of weight p, it should be of the form aibb. But aibb · e1 = ai (ap + a2p + · · · + a(p−1)p )bb 6= aibb which implies aibb 6∈ I1 , for any 1 ≤ i. Therefore, the mininum weight of I1 is 2p. The mininum weight of the codes I2 and I3 is a consequence of the proof of the next proposition. Proposition 3.2: The (inequivalent) minimal codes I2 and I3 of Proposition 3.1 have the same weight distribution. Proof: An F2 -basis for the code I2 is ˆ −b ˆ−a β = {µi = (G a)bi = G ˆbi |1 ≤ i ≤ p − 1}. For 1 ≤ i 6= j ≤ p − 1, we have supp(ˆ abi ) ∩ supp(ˆ abj ) = ∅. Hence, for an element α ∈ I2 , we have: Case 1. α is a sum of an even number of µi ’s. Thus i1

b−a α = µi1 + · · · + µi2k = 2k G ˆ(b + · · · + b

i2k

),

2 which isan element  of weight 2kp . Besides, in I2 we have p−1 at least distinct elements with weight 2kp2 . 2k

Case 2. α is a sum of an odd number of µi ’s. Thus, for k 0 ≥ 1, ˆ−a α = µi1 + · · · + µi2k0 −1 = G ˆ(bi1 + · · · + bi2k0 −1 ), which is an element of weight p3 − (2k 0 − 1)p2 = p2 (p − 2k 0 + 1) = 2kp2   p−1 p+1−2k 0 ). Hence, in I2 , there are = (where k = 2 2k 0 − 1   p−1 distinct elements with weight 2kp2 . p − 2k   p−1 Therefore, for each k ≥ 1, there are + 2k     p−1 p = elements of weight 2kp2 in I2 . p − 2k 2k

0

Notice that Proposition 3.2 actually exhibits a counterexample for Theorem B. In the following proposition, we study the minimal codes in F2 (Cpn × Cp ), for an odd prime p and n ≥ 3. Its proof is similar to the proof of Proposition 3.1. This gives a whole family of counterexamples to Theorem A. Proposition 3.3: Let n ≥ 3 be a positive integer and p an odd prime such that ¯2 generates U (Zp2 ) and G = hai × hbi be an abelian group, with o(a) = pn and o(b) = p. Then the minimal codes of F2 G are described in the following table. Code b I0 = hb ab bi = hGi p \ b I1 = hha i × hbi − Gi c i b I1i = hab − Gi i = 0, . . . , p − 1

2 I2 = h ap\ × hbi − hap\ i × hbii

For 1 ≤ i 6= j ≤ p − 1, supp(abpbbai ) ∩ supp(abpbbaj ) = ∅. Hence, for an element α ∈ I3 , we have:

p bi − hap \ I2i = had i × hbii i = 1, . . . , p − 1 ... 

k

\ Ik = h ap\ × hbi − apk−1 × hbii

k−1  k−1 i \ \ p p × hbii Iki = ha b − a i = 1, . . . , p − 1 ...  \ c − apn−2 In−1 = hhbi × hbii

 \ pn−1 bi − apn−2 In−1,i = ha\ × hbii i = 1, . . . , p − 1

Case 1. α is a sum of an even number of δi ’s. Thus ˆ − abpbb(ai1 + · · · + ai2k ), = 2k G

2 which isan element  of weight 2kp . Besides, in I3 we have p−1 at least distinct such elements with weight 2kp2 . 2k

Case 2. α is a sum of an odd number of δi ’s. Thus, for k 0 ≥ 1,

which is an element of weight p3 − (2k 0 − 1)p2 = p2 (p − 2k 0 + 1) = 2kp2 ,



Observe that the group G of Proposition 3.1 has exponent p2 and τ (p2 ) = 3, however, F2 G has four inequivalent minimal codes. This is a counterexample for Theorem A.

b − abpbbai |1 ≤ i ≤ p − 1}. γ = {δi = e3 ai = G

b − abpbb(ai1 + · · · + ai2k0 −1 ), α = δi1 + · · · + δi2k0 −1 = G

p−1 2k 0 − 1

Hence, in I3 , there are where k = =   p−1 distinct elements with weight 2kp2 . p − 2k   p−1 Therefore, also in I3 , for each k ≥ 1, there are + 2k     p−1 p = elements of weight 2kp2 . p − 2k 2k  (p−1)/2  X p As = 2p−1 , this proves that the weight 2k k=1 distribution of I2 and I3 are the same, but I2 and I3 are not equivalent. Besides, the mininum weight of these codes is 2p2 .

Similarly, an F2 -basis for the code I3 is

α = δi1 + · · · + δi2k



p+1−2k . 2

Dimension

Mininum Weight

1 p−1 p−1

pn+1 2pn 2pn

p(p − 1)

2pn−1

p(p − 1)

2pn−1

... pk−1 (p − 1)

2pn−k+1

pk−1 (p − 1)

2pn−k+1

... (n−1)

(p − 1)

2p

p(n−1) (p − 1)

2p

p

There are 2n inequivalent minimal codes in F2 (Cpn × Cp ). The following table presents the correspondence between the classes of G-isomorphisms of subgroups and the classes

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of equivalence of minimal codes in F2 (Cpn × Cp ). Rep. Subgroups Class G hai hap i × hbi p D haE bi ap

2

× hbi . . . D k E ap b D k+1 E ap × hbi ... hbi

By Lemma 4.1 and Proposition 4.2, the correspondence between classes of G-isomorphic subgroups of G = Cpn ×Cpn and classes of equivalent minimal codes of F2 (Cpn × Cpn ) is as follows:

Rep. Codes Class b I0 = hGi b I11 = hb a − Gi p \ b I1 = hha i × hbi − Gi p p \ c I21 = ha b − ha i × hbii

2 × hbi − hap\ i × hbii I2 = h ap\ ...

k pk b − ap\ I = had × hbii

Rep. Subgroups Class G p ha i D 2 E× hbi ap × hbi D .E. .

k+1,1

Ik+1 =

ap

k hbi − ap\ × hbii ... −1 i × hbii = hb b − hapn\

\ hhapk+1 i×

In−1

Now we present the minimal codes in F2 (Cpn × Cpn ), for an odd prime p and an integer n ≥ 1. This case exhibits a particular family for which Theorem A does hold. Again we shall omit the proof of some group-theoretical results that are used. Lemma 4.1: Let n ≥ 1 be an integer and p be an odd prime such that ¯ 2 generates U (Zp2 ) and G = hai × hbi be an abelian group isomorphic to Cpn ×Cpn of order p2n . Then the subgroups that give rise to primitive idempotents in the group i i algebra F2 G are hap i × hbi (or hai × hbp i), for i = 0, . . . , n, and any other G-isomorphic to any of these. Proposition 4.2: Let n ≥ 1 be a positive integer and p be an odd prime such that ¯ 2 generates U (Zp2 ) and G = hai × hbi be an abelian group with o(a) = o(b) = pn . Then the minimal codes of F2 G are described in the following table. Dimension

Minimum Weight

1 p−1 p−1

p2n 2p2n−1 2p2n−1

b hb ab bi = hGi b hhap\ i × hbi − Gi i \ p b hhab i × hb i − Gi i = 0...,p − 1 ... 

k

\ × hbi − apk−1 × hbii h ap\

p

(p − 1)

2p2n−k

k = 1, . . . , n − 1



 hhabi \ i × bpk − habi i \ × bpk−1 i

pk−1 (p − 1)

2p2n−k

i = 1, . . . , p − 1 ...  \ c − apn−1 hhbi × hbii 

\ [ i i hhab i − hab i × bpn−1 i i = 1, . . . , p − 1

2pn

p(n−1) (p − 1)

2pn

It is known that if a product of two primitive idempotents is not primitive, then it splits as a sum of primitive idempotents where the number of summands depends on many parameters, including the order of the group and the size of the field. As we have seen in the examples shown above, under suitable assumptions on the order of the group, we may have a good control on the number of such summands. We worked with abelian p-groups which are direct product of only two cyclic factors. It seems that one needs to work much harder to find a general expression for the primitive idempotents for group codes of a general abelian group, even in the binary situation. It is important to note that the classes of G-isomorphisms of subgroups of G play an interesting role in the classification of the minimal equivalent codes. Also, the use of subgroups allows us to express the idempotents in a much simpler way if we compare their expression with their analogues using the polynomial approach. The authors would like to thank FAPEMIG, FAPESP and PROCAD/CAPES for the financial support for this project. The first author would like to thank the hospitality of IMEUSP during the realization of much of this research. R EFERENCES

... p(n−1) (p − 1)

V. C ONCLUSION

ACKNOWLEDGMENT

... k−1

× hbi ...

hbi

IV. A POSITIVE RESULT

Code

k

Rep. Codes Class b I0 = hGi p \ b I1 = hha i × hbi − Gi

\  2 I2 = h ap × hbi − hap\ i × hbii ...

k

\  Ik = h ap\ × hbi − apk−1 × hbii ...  \ c − apn−1 In = hhbi × hbii

There are τ (pn ) = n + 1 inequivalent minimal codes in F2 (Cpn × Cpn ). Proof: The idempotents follow from Theorem 1.1 and Lemma 4.1. Dimension and minimum weights are computed as in [3, Section 5].

[1] S.D. Berman, Semisimple cyclic and abelian codes, II, Kybernetika, 3, (1967) 21-30. [2] I.F. Blake and R.C. Mullin, The Mathematical Theory of Coding, New York, USA: Academic Press, 1975. [3] R.A. Ferraz and C. Polcino Milies, Idempotents in group algebras and minimal abelian codes, Finite Fields and their Applications, 13, (2007) 382-393. [4] C.J. Hillar and D.L. Rhea, Automorphisms of finite abelian groups, American Math. Monthly 114 n. 10 (2007) 917-923. [5] F.J. MacWilliams,Binary codes which are ideals in the group algebra of an abelian group, Bell System Tech. Journal, 44, (1970) 987-1011. [6] R.L. Miller, Minimal codes in abelian group algebras, Journal of Combinatorial Theory, Series A, 26 (1979) 166-178. [7] C. Polcino Milies, S.K. Sehgal, An Introduction to Group Rings, Dordrecht, Netherlands: Kluwer Academic Publishers, 2002.

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