... More recently, as coding theory ... consider from the view of coding theo
Codes, Lattices and Modular Forms YoungJu Choie Department of Mathematics Pohang University of Science and Technology Pohang, 790-784, Korea Email:
[email protected] Steven T. Dougherty Department of Mathematics University of Scranton Scranton, PA 18510, USA Email:
[email protected] June 22, 2011 Abstract We describe various constructions of unimodular lattices from codes over finite rings and modular forms constructed from those lattices.
1
Introduction
The first connection between self-dual codes and lattices was made between binary codes and real unimodular lattices, see [11] for a description of this early work. Several connections were made between ternary and quaternary self-dual codes and complex unimodular lattices which were not as natural as the original connection. More recently, as coding theory expanded to consider codes over finite rings and not just codes over fields, a canonical connection was discovered between codes over the ring Z2m and real unimodular lattices, see [1]. Additionally, this connection was extended to construct modular forms from the given lattices. Following this work a great many papers were written describing many results which flowed naturally over these canonically bridges between codes, lattices and modular forms. Recently, another canonical bridge was described in the papers [3] and [4] (both in review) between codes over the rings S2m and complex unimodular lattices. This bridge again extends naturally into the realm of modular forms, in this case Hermitian 1
modular forms. At present another bridge is being studied connecting codes over the rings Σ2m and Quaternionic unimodular lattices and their associated modular forms, see [5] (in preparation). Perhaps as interesting as the connection between the codes and the lattices is the fact that these rings have canonical gray maps which map into Z2m which extend over the whole space. The gray map is naturally associated to the gray maps between the rings of order 4 and the field of order 2. These maps make a bridge between self-dual codes over S2m and Σ2m and self-dual codes over Z2m . When determining which rings are important to consider from the view of coding theory, it is necessary to find a ring such that the naturally defined weights correspond to the Euclidean weights via a gray map and to the norms via the natural lattice construction. In this work, we shall give a unified presentation of the major results of these works.
2
Codes
A code of length n over a ring A is a subset of An . If it is a submodule then we say the code is linear. In case the ring is not commutative, we shall assume that all multiplication is on the left and that the code is linear if it is a left module.
2.1
Rings
The first family of rings we consider is the ring Z2m = Z/2mZ. The Euclidean weight of a P vector w = (wi ) ∈ Zn2m is given by wt(w) = min{(wi )2 , (2m − wi )2 }. The second family of rings are the rings S2m := Z2m + ωZ2m = Z[i]/2mZ[i] by S2m . The element ω corresponds to 1 + i, and we equip each ring with an involution a + bω = a + bω, a, b ∈ Z. We have that |S2m | = (2m)2 , ωω = 2, w2 = 2ω + (2m − 2) ω = 2 + (2m − 1)ω, and a + bω = (a + 2b) + (2m − 1)bω. The third family of rings are the rings Σ2m which are given by Z2m + αZ2m + βZ2m + γZ2m = Z[i, j, k]/2mZ[i, j, k], by Σ2m , where i, j, k are elements of the Quaternions H. The element α corresponds to 1 + i, the element β corresponds to 1 + j, and the element γ corresponds to 1 + k. We equip each ring with an involution a + bα + cβ + dγ = a + bα + cβ + dγ. We have that |Σ2m | = (2m)4 , αα = ββ = γγ = 2, α2 = 2α −2, β 2 = 2β −2, γ 2 = 2γ −2, α = 2 − α, β = 2 − β, γ = 2 − γ, αβ = βγ = γα = α + β + γ − 2, and βα = α + β − γ, γβ = −α+β +γ, αγ = α−β +γ. It follows that a + bα + cγ + dγ = a+2b+2c+2d−bα−cβ −dγ. For a code over the ring A we define the complete weight enumerator by (1)
WC (Xa ) =
X Y c∈C a∈A
2
xna a (c)
where na (c) is the number of times a appears in the codeword c. The complete joint weight enumerator is defined by JC1 ,C2 ,..,Cg (Xa ) =
(2)
X
X
X
...
c1 ∈C1 c2 ∈C2
xna a (c1 ,c2 ,...cg )
cg ∈Cg
where na (c1 , c2 , . . . cg ) is |{i | (c1i , c2i , . . . cg i = a}|, for a ∈ Ag . The MacWilliams relations for these weight enumerators for codes over any ring can be found in [6] and [7].
2.2
Gray Maps
We shall describe two families of gray maps connecting the rings S2m and Σ2m with Z2m such that the natural weights over these rings are connected to the Euclidean weight via the gray map. The gray map for the ring S2m is Φ2m : S2m → Z22m . It is given by (3)
Φ(a + bω) = (b, a + b).
This map is a generalization of the map given for codes over F2 + uF2 in [8]. The gray map for the ring Σ2m is Ψ2m : Σ2m → Z42m . It is given by Ψ2m (a + bα + bγ + cδ) = (b, c, d, a + b + c + d). Both maps extended to the whole space An by applying it coordinatewise. The maps Φ2m and Ψ2m are linear. The proofs of the following appear in [4] and [5]. Theorem 2.1 If C is a self-dual code of length n over S2m , then Φ2m (C) is a self-dual code of length 2n over Z2m . If C is a self-dual code of length n over Σ2m , then Ψ2m (C) is a self-dual code of length 4n over Z2m .
3
Lattices
As usual we denote the integers by Z, the Gaussian integers Z[i] by G, and Z[i, j, k] by O. A lattice in Rn is a free Z- module, a lattice in Cn is a free G-module and a lattice in Hn is a free O-module. The standard inner product is attached to the ambient spaces: (4)
v·u=
X
vi ui
where x is the usual conjugation for C and H and the identity for R. For any lattice L in the ambient space K n with ring of integers B, we define L∗ = {u ∈ K n | u · v ∈ B for all v ∈ L}. A vector v has norm N (v) = v · v. If the norm of every vector in a unimodular lattice is even then we say it is an even lattice. An integral lattice has L ⊆ L∗ and a unimodular lattice has L = L∗ . 3
3.1
Constructions of Lattices
Construction A for building real lattice from codes is first given in [2]. The function ρ maps Z2k to Z sending 0, 1, . . . , k to 0, 1, . . . , k and k + 1, . . . , 2k − 1 to 1 − k, . . . , −1, respectively. It is shown in [1] that if C is a self-dual code of length n over Z2k , then 1 ΛR (C) = √ {ρ(C) + 2kZn }, 2k is an n-dimensional unimodular lattice, with ρ applied coordinatewise to the codeword. The lattice has minimum norm min{2k, dE /2k} where dE is the minimum Euclidean weight of C and ΛR (C) is an even unimodular lattice if C is Type II. For complex lattices the following construction was given in [4]. Denote the reduction map modulo 2m by:
(5)
n h˜C : G n → S2m .
−1 This reduction is a group homomorphism and h˜C (C), the preimage of a code C defined over S2m , is a free G-module. The induced lattice is defined as follows:
(6)
1 −1 ΛC (C) := √ h˜C (C) = {v ∈ G n | v 2m
(mod 2mG) ∈ C},
where ω corresponds to 1 + i. For Quaternionic lattices the following construction is in [5]. Denote the reduction map modulo 2m by:
(7)
h˜H : On → Σn2m .
−1 It is a group homomorphism and h˜H (C), the preimage of a code C defined over Σ2m , is a free O-module. The lattice induced from a code C is defined as follows: 1 −1 (8) ΛH (C) := √ h˜H (C) = {v ∈ On | v (mod 2mO) ∈ C}, 2m
where α corresponds to 1 + i β corresponds to 1 + j and γ corresponds to 1 + k. The following appear in [1], [4] and [5]. We denote the minimum Euclidean weight of a code C by dE (C). Theorem 3.1 If C is a self-dual code over Z2m then ΛR (C) is a real unimodular lattice and (C) ΛR (C) is even if C is Type II. The minimum norm of the lattice is min{ dE2m , 2m}. If C is a self-dual code over S2m then ΛC (C) is a complex unimodular lattice and ΛC (C) is even (C) if C is Type II. The minimum norm of the lattice is min{ dE2m , 2m}. If C is a self-dual code over Σ2m then ΛH (C) is a Quaternionic unimodular lattice, and ΛH (C) is even if C is dE Type II. The minimum norm of the lattice is min{ 2m , 2m}. 4
The relationship between the codes and lattices can be easily seen in the following commutative diagrams. Hxn
ΛH
−−−−−→ Rx4n
Cxn
ψ
ΛR
Σn2m −−−−−→ Z42m
φ
ΛR
n S2m −−−−−→ Z22m
Ψ2m
4
−−−−−→ Rx2n
ΛC
Φ2m
Hermitian Jacobi forms and Modular forms of genus g
We recall the definitions of Hermitian Jacobi forms and theta-functions. We follow the definitions given in [9]. Let τ − τ∗ Hg := {τ ∈ M2g×2g (C) | > 0}. 2i Here τ ∗ = t τ . The Hermitian symplectic group ∗
SPg (C) := {M ∈ M2g×2g (C) | M JM = J}, J =
0 −1g 1g 0
!
,
acts on Hg in the usual way. The Hermitian modular group of genus g associated with G is defined by Γg (G) := Spg (C) ∩ M2g×2g (G). Definition 1 A holomorphic function f : H1 × C2 → C is said to be a Hermitian Jacobi form of weight k and index m with respect to G if it satisfies 1. cz1 z2
(f |k,m M )(τ, z1 , z2 ) := (cτ + d)−k e−2πim cτ +d f (M τ, = f (τ, z1 , z2 ), ∀M =
∗ ∗ c d
z1 z2 , ) cτ + d cτ + d
!
∈ Γ1 (G),
2. (f |m [λ, µ])(τ, z1 , z2 ) := e2πim(N (λ)τ +λz1 +λz2 ) f (τ, z1 + λτ + µ, z2 + λτ + µ). It has the following Fourier expansion: 3. f (τ, z1 , z2 ) =
∞ X
X
n=0 t∈G,N (t)≤4mn
5
c(n, t)e2πi(nτ +tz1 +tz2 ) .
The following theorem, which is in [4], gives a connection between a theta series defined over the lattices induced from codes and their complete weight enumerators. Theorem 4.1 Let C be a code over S2m . Let ΛC (C) be a lattice induced from the code C over S2m . From the complete weight enumerator cweC (X0 , .., X` ||S2m | = m), one constructs the theta-series ΘΛC (C) associated with ΛC (C): (9)
ΘΛC (C), √2m
2m
(1,..,1) (τ, z1 , z2 )
= cweC (θm,µ (τ, z1 , z2 ) | µ ∈ S2m ),
Here, for each µ ∈ S2m , {θm,µ } is given by X
(10) θm,µ (τ, z1 , z2 ) := r∈O,r≡µ
q
N (r) 4m
r
r
ξ12 ξ22 , q = e2πiτ , ξ1 = e2πiz1 , ξ2 = e2πiz2 .
(mod 2mG)
The following appears in [4] and relates Type II codes over S2m and Hermitian Jacobi forms. Theorem 4.2 Let C be a Type II code of length n over S2m . Then cweC (θm,µ (τ, z1 , z2 ) | µ ∈ S2m ) is a Hermitian Jacobi form of weight n and index mn. Definition 2 A holomorphic function F : Hg → C is called a Hermitian modular form of weight k of genus g if −k
F (M τ ) = Det(Cτ + C) F (τ ), ∀M =
∗ ∗ C D
!
∈ Γg ,
with a proper holomorphic condition at each cusp in the case of g = 1. The following theorems appear in [4]. Theorem 4.3 Let Cj , 1 ≤ j ≤ g, be a code of length n over S2m and Λj be an induced lattice from the code Cj , i.e., Λj = ΛC (Cj ). Let JC1 ,C2 ,..,Cg (X) be the complete joint weight √ enumerator of the codes Cj , 1 ≤ j ≤ g. Then the following holds for Y = 2m = √1 (2m, 2m, .., 2m)t : 2m (11)
g (g) ΘΛ1 ,Λ2 ,..,Λg ;Y (τ, z1 , z2 ) = JC1 ,C2 ,..,Cg (θm,µ (τ, z1 , z2 ) | µ ∈ S2m ).
Theorem 4.4 Let Cj , 1 ≤ j ≤ g, be a Type II code of length n over S2m . Let JC1 ,C2 ,..,Cg (X) be the complete joint weight enumerator of the codes Cj , 1 ≤ j ≤ g. Then g (g) JC1 ,C2 ,...,Cg (θm,µ (τ, 0, 0) | µ ∈ S2m )
is a Hermitian modular form of weight n and genus g. 6
5
Jacobi forms on Half-Spaces of Quaternions
In this section we extend this idea by studying the connection between weight enumerators of codes over the Quaternionic ring Σ2m and modular forms over the Half space of Quaternions. More precisely, the Theta series formed from the complete weight enumerators of the codes over Σ2m is a modular form over the Half space of Quaternion. Also modular forms of higher genus have been derived from the joint weight enumerators of codes over Σ2m . The Jacobi group of Quaternion H will be denoted by ΓJ (H) := SL2 (O) ∝ O2 . This group acts on H × H, where H denotes the ! complex upper half plane. The action of α β ΓJ (H) on the space H × H is given by, ∈ SL2 (OH ), (τ, z) ∈ H × H, γ δ α β γ δ 2 and, for all [λ, µ] ∈ OH ,
!
· (τ, z) := (
ατ + β 1 , z) γτ + δ γτ + β
[λ, µ] · (τ, z) := (τ, z + λτ + µ).
Definition 3 Given k ∈ 21 Z and m ∈ O, a function f : H × H → C is said to be a Jacobi forms of weight k and index m on H if it is analytic function satisfying 1. czz
(f |k,m M )(τ, z) := (cτ + d)−k e−2πi(m cτ +d ) f (M · (τ, z)) = f (τ, z), ∀M =
∗ ∗ c d
!
∈ SL2 (O).
2. (f |m [λ, µ])(τ, z) := e−2πi(m(λλτ +λz+zλ))) f (τ, [λ, µ] · z) = f (τ, z). And it has the following Fourier expansion: 3. f (τ, z) =
c(`, λ)e2πi`τ e2πi(λz+zλ) .
X
` ∈ N, λ ∈ O∗ 2` ≥ λλ Here O∗ is the dual of O.
7
The following theorem gives a connection between a theta series defined over the lattices induced from codes and their complete weight enumerators and appears in [5] which is in preparation. Theorem 5.1 Let C be a code over Σ2m . Let ΛH (C) be a lattice induced from C over Σ2m . From the complete weight enumerator cweC (X0 , .., X` ||Σ2m | = `), one constructs the following theta-series associated with ΛH (C): Let (12)
ΘO,Y (τ, z) :=
X
e2πi
x·xτ 2
+
(x·Y )z+z(x·Y ) 2
.
x∈O
Then (13)
ΘΛ(C),2m(1,..,1) (τ, z) = cweC (θH,2m,µ (τ, z) | µ ∈ Σ2m ),
where {θH,2m,µ } is given, for each µ, (14)
θH,2m,µ (τ, z) :=
rrτ
e2πi 4m e2πi
X r∈O,r≡µ
rz+zr 2m
.
(mod (2m))
Theorem 5.2 Let C be a Type II code of length n over Σ2m . Then cweC (θH,2m,µ (τ, z) | µ ∈ Σ2m ) is a Quaternion Jacobi form of weight 2n and index n. Next we consider modular forms of higher genus. Let Hg (H) = {Z = X + iY ; X, Y ∈ M at(g × g, H), X = X t , Y = Y t > 0}. The Quaternionic modular group of degree g is given by {M ∈ M at(2g, O) | M t JM = J, J =
0 1 −1 0
!
}.
This group acts on Hg (H) as −1
M · Z = (AZ + B)(CZ + D) , M =
A B C D
!
∈ Γg (H), Z ∈ Hg (H).
Let us use the following standard notation; for matrices A, B ∈ M at(` × m, H) define σ(A, B) =
trace(AB ∗ +BA∗) . 2
Here A∗ denotes the transpose of a matrix A. 8
Definition 4 (Modular Form of degree g over Γg (O)) A holomorphic function f : Hg (H) → C is said to be a modular forms of degree g with weight k, k ≡ 0 (mod 2), on Γn () if it satisfies (f |k M )(Z) := Det(Cτ + D)−k f (M · Z) = f (Z), ∀M =
5.1
∗ ∗ C D
!
∈ Γg (O), One has to add the condition of boundedness in the case g = 1.
Construction of Quaternionic Modular forms of genus g over the Quaternions
In this section, we consider a higher genus Quaternionic modular form and derive a connection with the joint weight enumerators of codes over Σ2m . (g) Consider, for each µ ∈ Σg , the Theta-function θH,2m,µ : Hg (H) × Hg (H) → C; (15)
(g)
θH,2m,µ (τ, z) =
e2πi
X r∈Og ,r≡µ
r ∗ ·τ ·r 4m
e2πi
r ∗ ·z+z ∗ ·r 2
.
(mod (2mO)g )
For given lattices Λ1 , .., Λg , and for each fixed Y ∈ Λ1 ∩ .. ∩ Λg , let us consider the following theta-series ΘΛ1 ,..,Λg ;Y : Hg × Hg → C defined as: (16)
ΘH,Λ1 ,..,Λg ;Y (τ, z) =
e2πi
X
T r(x∗ ·τ ·x) 4m
e2πi
(xY )∗ ·z+z ∗ ·(xY ) 2
.
x∈Λ1 ×Λ2 ×..×Λg
The next theorems which appear in [5] state a connection between the theta-series defined over the lattices induced from codes and their joint weight enumerators. Theorem 5.3 Let Cj , 1 ≤ j ≤ g, be a code of length n over Σ2m and Λj = ΛH (Cj ) be the induced lattice from the code. Let JC1 ,C2 ,..,Cg (X) be the complete joint weight enumerator of √ 1 the codes Cj , 1 ≤ j ≤ g. Then the following holds for Y = 2m = √2m (2m, 2m, .., 2m)t : (17)
(g) ΘH,Λ1 ,Λ2 ,..,Λg ;Y (τ, z) = JC1 ,C2 ,..,Cg (θm,µ (τ, z) | µ ∈ Σg2m ).
Theorem 5.4 Let Cj , 1 ≤ j ≤ g, be a Type II code of length n over Σ2m . Let JC1 ,C2 ,..,Cg (X) be the complete joint weight enumerator of the codes Cj , 1 ≤ j ≤ g. Then (g)
JC1 ,C2 ,...,Cg (θH,2m,µ (τ, 0) | µ ∈ Σg2m ) is a Quaternionic modular form of weight n and genus g.
9
References [1] E. Bannai, S.T. Dougherty, M. Harada, and M. Oura, Type II Codes, Even Unimodular Lattices, and Invariant Rings, IEEE-IT, Vol. 45, No. 4, 1999, 1194-1205. [2] J.H. Conway and N.J.A. Sloane, Sphere Packing, Lattices and Groups (2nd ed.), Springer-Verlag, New York, 1993. [3] Y. Choie and S.T. Dougherty, Codes over Z4 + ωZ4 , Complex Lattices and Hermitian Modular Forms, submitted. [4] Y. Choie and S.T. Dougherty, Codes over Z2m +ωZ2m , Complex Lattices and Hermitian Modular Forms, submitted. [5] Y. Choie and S.T. Dougherty, Codes over Σ2m , Quaternionic Lattices and Modular Forms , in preparation. [6] MacWilliams Relations for Codes over http://academic.uofs.edu/faculty.doughertys1/publ.htm.
Groups
and
Rings,
[7] MacWilliams Relations for Joint Weight Enumerators for Codes over Rings http://academic.uofs.edu/faculty.doughertys1/publ.htm. [8] S.T. Dougherty, P. Gaborit, M. Harada, and P. Sol´e, Type II codes over F2 + uF2 , IEEE Trans. Inform. Theory, Vol 45, No. 1, 1999, 32-45. [9] V.A. Gritsenko, The effect of modular operators on the Fourier-Jacobi coefficients of modular forms, Math. Sb. 119 (161), No. 2, 1982, 248-277. [10] K. Haverkamp, Hermitian Jacobi forms, Results in Math. Vol. 29, 1996, 78-89. [11] E.Rains and N.J.A. Sloane, Self-dual codes, in the Handbook of Coding Theory, V.S. Pless and W.C. Huffman, eds., Elsevier, Amsterdam, 1998, 177-294.
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