Weight Enumerators Of Self-dual Codes Research Articles

Symmetries of weight enumerators and applications to Reed-Muller codes

Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.

Byte weight enumerators of codes over 𝔽<SUB align="right">p and modular forms over a totally real field

In this paper, we obtain Jacobi forms over kp from byte weight enumerators of self-dual codes over Fp, where p is an odd prime, kp is the totally real field of the pth cyclotomic field and Fp is the finite field of order p. We also determine Siegel modular forms of genus g (g ≥ 1 is an integer) over kp by substituting certain theta series into byte weight enumerators in genus g of self-dual codes over Fp for all p ∈ P, where the set P consists of all those odd primes p for which the ring of algebraic integers of kp is a Euclidean domain. Further, we define some partial Epstein zeta functions and derive their functional equation using the Mellin transform of the theta series.

On the Existence of Certain Optimal Self-Dual Codes with Lengths Between 74 and 116

The existence of optimal binary self-dual codes is a long-standing research problem. In this paper, we present some results concerning the decomposition of binary self-dual codes with a dihedral automorphism group $D_{2p}$, where $p$ is a prime. These results are applied to construct new self-dual codes with length $78$ or $116$. We obtain $16$ inequivalent self-dual $[78,39,14]$ codes, four of which have new weight enumerators. We also show that there are at least $141$ inequivalent self-dual $[116,58,18]$ codes, most of which are new up to equivalence. Meanwhile, we give some restrictions on the weight enumerators of singly even self-dual codes. We use these restrictions to exclude some possible weight enumerators of self-dual codes with lengths $74$, $76$, $82$, $98$ and $100$.

Lee weight enumerators of self-dual codes and theta functions

The theory of modular forms, in particular theta functions, and coding theory are in a remarkable way connected. The connection is established by defining a suitable lattice corresponding to the given code, and considering its theta function. First we define some special theta functions, and determine transformation formulas. Then it is proved that the theta function of the lattice corresponding to the code can be expressed in terms of the Lee weight enumerator. In particular if the code is self-dual and the length is a multiple of $8$ this theta function is a modular form for some subgroup of the modular group. Using the known structure of this space of modular forms we can derive linear relations between the coefficients of the Lee weight enumerator. And from these relations we can get an upper bound for the minimal Lee distance of self-dual $p$-ary codes.

Codes over [formula omitted], Jacobi forms and Hilbert–Siegel modular forms over [formula omitted

We study codes over a finite field F 4 . We relate self-dual codes over F 4 to real 5-modular lattices and to self-dual codes over F 2 via a Gray map. We construct Jacobi forms over Q ( 5 ) from the complete weight enumerators of self-dual codes over F 4 . Furthermore, we relate Hilbert–Siegel forms to the joint weight enumerators of self-dual codes over F 4 .

Note on the g-fold Joint Weight Enumerators of Self-Dual Codes over ℤ k

Recently there has been interest in self-dual codes over finite rings. In this note, g-fold joint weight enumerators of codes over the ring ℤ k of integers modulo k are introduced as a generalization of the biweight enumerators. We establish the MacWilliams relations for these weight enumerators and investigate the biweight enumerators of self-dual codes over ℤ k . We derive Gleason-type theorems for the corresponding biweight enumerators with the help of invariant theory.

On the Hamming Weight Enumerators of Self-Dual Codes over [formula omitted

In this article, we investigate the Hamming weight enumerators of self-dual codes over Fqand Zk. Using invariant theory, we determine a basis for the space of invariants to which the Hamming weight enumerators belong for self-dual codes over Fqand Zk.

Double Circulant Codes over Z\!Z_{\bf 4} and Even Unimodular Lattices

With the help of some new results about weight enumerators of self-dual codes over $$\mathbb{Z}_4 $$ we investigate a class of double circulant codes over $$\mathbb{Z}_4 $$ , one of which leads to an extremal even unimodular 40–dimensional lattice. It is conjectured that there should be “Nine more constructions of the Leech lattice”

Weight enumerators of self-dual codes

Some construction techniques for self-dual codes are investigated, and the authors construct a singly-even self-dual (48,24,10)-code with a weight enumerator that was not known to be attainable. It is shown that there exists a singly-even self-dual code C' of length n=48 and minimum weight d=10 whose weight enumerator is prescribed in the work of J.H. Conway et al. (see ibid., vol.36, no.5, p.1319-33, 1990). Two self-dual codes of length n are called neighbors, provided their intersection is a code of dimension (n/2)-1. The code C' is a neighbor of the extended quadratic residue code of length 48.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Generalizations of Gleason's theorem on weight enumerators of self-dual codes

Gleason has recently shown that the weight enumerators of binary and ternary self-dual codes are polynomials in two given polynomials. In this paper it is shown that classical invariant theory permits a straightforward and systematic proof of Gleason's theorems and their generalizations. The joint weight enumerator of two codes (analogous to the joint density function of two random variables) is defined and shown to satisfy a MacWilliams theorem. Invariant theory is then applied to generalize Gleason's theorem to the complete weight enumerator of self-dual codes over GF(3) , the Lee metric enumerator over GF(5) (given by Klein in 1884!) and over GF(7) (given by Maschke in 1893!), the Hamming enumerator over GF(q) , and over GF(4) with all weights divisible by 2, the joint enumerator of two self-dual codes over GF(2) , and a number of other results.