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Codes over rings of size four, Hermitian lattices, and correspondingtheta functions
Abstract
Let be an imaginary quadratic field with ring of integers
\O_K, where is a square free integer such that
and be a linear code defined over \O_K/2\O_K. The level
theta function \Th_{\L_{\ell} (C)} of C is defined on the lattice
\L_{\ell} (C):= \set {x \in \O_K^n : \rho_\ell (x) \in C}, where
\rho_{\ell}:\O_K \rightarrow \O_K/2\O_K is the natural projection. In this
paper, we prove that: % i) for any such that , \Th_{\Lambda_\ell}(q) and \Th_{\Lambda_{\ell^\prime}}(q) have
the same coefficients up to , % ii) for , \Th_{\L_{\ell}} (C) determines the code C uniquely, %
iii) for there is a positive dimensional
family of symmetrized weight enumerator polynomials corresponding to
\Th_{\La_\ell}(C).
... For examples of non-equivalent codes corresponding to the same weight enumerator the reader can check [13], [14], and [12] ...
This is a survey on weight enumerators, zeta functions and Riemann hypothesis for linear codes.
... Such curves have been of much interest lately because of their use in many theoretical and applicative situations. The first part of the paper is based on several papers on the topic of genus two curves with split Jacobians; see [1,[3][4][5][6][7][8][9][11][12][13][14][16][17][18][19][20][21] among others. ...
We study the singular locus on the algebraic surface of genus 2 curves
with a -split Jacobian. Such surface was computed by Shaska in
\cite{deg3} for n=3, and Shaska at al. in \cite{deg5} for n=5. We show that
the singular locus for n=2 is exactly th locus of the curves of automorphism
group or . For n=3 we use a birational parametrization of the
surface discovered in \cite{deg3} to show that the singular locus is a
0-dimensional subvariety consisting exactly of three genus 2 curves (up to
isomorphism) which have automorphism group or . We further show that
the birational parametrization used in would work for all if
is a rational surface.
Using codes defined over F 4 and F 2 × F 2 , we simultaneously define the theta series of corresponding lattices for both real and imaginary quadratic fields Q ( d ) with d ≡ 1 mod 4 a square-free integer. For such a code, we use its weight enumerator to prove which term in the code’s corresponding theta series is the first to depend on the choice of d . For a given choice of real or imaginary quadratic field, we find conditions on the length of the code relative to the choice of quadratic field. When these conditions are satisfied, the generated theta series is unique to the code’s symmetric weight enumerator. We show that whilst these conditions ensure all non-equivalent codes will produce distinct theta series, for other codes that do not satisfy this condition, the length of the code and choice of quadratic field is not always enough to determine if the corresponding theta series will be unique.
Let with square-free and let denote the ring of integers of K. Let be a linear code where is if and if . One has a surjective ring homomorphism given by reduction modulo . The inverse image is a lattice associated to the code . One can associate to a theta series . In this paper we consider how the theta series varies as one varies the value d. In particular, we show that for with , one has
We are interested in the -holomorphic automorphic functions on a
g-dimensional complex space endowed with a positive
definite hermitian form and associated to isotropic discrete subgroups
of rank . We show that they form an infinite reproducing kernel
Hilbert space which looks like a tensor product of a theta Fock-Bargmann space
on and the classical
Fock-Bargmann space on . Moreover, we provide an explicit
orthonormal basis using Fourier series and we give the expression of its
reproducing kernel function in terms of Riemann theta function of several
variables with special characteristics.
The purpose of this paper is to study a further connection between linear codes over three kinds of finite rings and Hermitian lattices over a complex quadratic field (Formula presented.), where l >0 is a square free integer such that l≡3(mod4). Shaska et al. (Finite Fields Appl 16(2): 75–87, 2010) consider a ring R = OKpOK (p is a prime) and study Hermitian lattices constructed from codes over the ring R. We consider a more general ring (Formula presented.), where e ≥1. Using pe allows us to make a connection from a code to a much larger family of lattices. That is, we are not restricted to those lattices whose minimum norm is less than p. We first show that R is isomorphic to one of the following three non-isomorphic rings: a Galois ring GR pe, 2)(Formula presented.), and (Formula presented.). We then prove that the theta functions of the Hermitian lattices constructed from codes over these three rings are determined by the complete weight enumerators of those codes. We show that self-dual codes over R produce unimodular Hermitian lattices. We also discuss the existence of Hermitian self-dual codes over R. Furthermore, we present MacWilliams’ relations for codes over R.
This paper is the first version of a project of classifying all superelliptic
curves of genus according to their automorphism group. We determine
the parametric equations in each family, the corresponding signature of the
group, the dimension of the family, and the inclussion among such families. At
a later stage it will be determined the decomposition of the Jacobians and each
locus in the moduli spaces of curves.
This paper is the first version of a project of classifying all su-perelliptic curves of genus g ≤ 48 according to their automorphism group. We determine the parametric equations in each family, the corresponding signature of the group, the dimension of the family, and the inclussion among such families. At a later stage it will be determined the decomposition of the Jacobians and each locus in the moduli spaces of curves.
This short note describes some of the papers and activities which took place during the NATO Advanced Study Institute in Vlora, Albania during April 28 - May 9, 2008.
Codes over an infinite family of rings which are an extension of the binary field are defined. Two Gray maps to the binary field are attached and are shown to be conjugate. Euclidean and Hermitian self-dual codes are related to binary self-dual and formally self-dual codes, giving a construction of formally self-dual codes from a collection of arbitrary binary codes. We relate codes over these rings to complex lattices. A Singleton bound is proved for these codes with respect to the Lee weight. The structure of cyclic codes and their Gray image is studied. Infinite families of self-dual and formally self-dual quasi-cyclic codes are constructed from these codes.
In this paper we continue the study of codes over imaginary quadratic fields and their weight enumerators and theta functions. We present new examples of non-equivalent codes over rings of characteristic p=2 and p=5 which have the same theta functions. We also look at a generalization of codes over imaginary quadratic fields, providing examples of non-equivalent pairs with the same theta function for p=3 and p=5.
Let \X be an algebraic curve of genus defined over a field
\F_q of characteristic . From \X, under certain conditions, we can
construct an algebraic geometry code C. If the code C is self-orthogonal
under the symplectic product then we can construct a quantum code Q, called a
QAG-code. In this paper we study the construction of such codes from curves
with automorphisms and the relation between the automorphism group of the curve
\X and the codes C and Q.
Let \chi be a character on a discrete subgroup \Gamma of rank one of the
additive group (C,+). We construct a complete orthonormal basis of the Hilbert
space of (L^2,\Gamma,\chi)-theta functions. Furthermore, we show that it
possesses a Hilbertian orthogonal decomposition involving the L^2-eigenspaces
of the Landau operator \Delta_\nu; \nu>0, associated to the eigenvalues \nu m.
For m=0, the associated L^2-eigenspace is the Hilbert subspace of entire
(L^2,\Gamma,\chi)-theta functions. Corresponding orthonormal basis are
constructed and the corresponding reproducing kernel can be expressed in terms
of the generalized theta function of characteristic [\alpha,0].
Let be a square-free integer congruent to 3 mod 4 and the ring of integers of the imaginary quadratic field K=Q(\sqrt{-\ekk})> Codes C over rings determine lattices over K. If then the ring is isomorphic to or . Given a code C over , theta functions on the corresponding lattices are defined. These theta series can be written in terms of the complete weight enumerators of C. We show that for any two the first terms of their corresponding theta functions are the same. Moreover, we conjecture that for there is a unique symmetric weight enumerator corresponding to a given theta function. We verify the conjecture for primes , and small n.
We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over GF(4) and F 2 × F 2, and give an interesting geometric characterization of the theta series that arise in terms of the basic strongly I modular lattice Z + √ℓZ. We identify some of the hermitian lattices constructed and observe an interesting pair of non-isomorphic 3/2 dimensional codes over F 2 × F 2 that give rise to isomorphic hermitian lattices when constructed at the lowest level 7 but nonisomorphic lattices at higher levels. The results show that the two alphabets GF(4) and F 2 × F 2 are complementary and raise the natural question as to whether there are other such complementary alphabets for codes.
This paper studies the relationship between error-correcting codes over GF(4) and complex lattices (more precisely, [ω]-modules in n, where ). The theta-functions of self-dual lattices are characterized. Two general methods are presented for constructing lattices from codes. Several examples are given, including a new lattice sphere-packing in 36.
A quadratic Jacobi identity to the septic base is introduced and proved by means of modular lattices and codes over rings. As an application the theta series of all the 6-dimensional 7-modular lattices with an Hermitian structure over Q(−7) are derived.
Sphere packings, lattices and groups, Second, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
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- N J A Sloane
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