Weight Enumerators Of Linear Codes Research Articles

Complete Weight Enumerators of Linear Codes Based on Weil Sums

With a proper defining set, we define a class of linear codes and determine their complete weight enumerators and weight enumerators using Weil sums. They only have two or three nonzero weights, and some of them are optimal with respect to the Griesmer bound or Markus Grassl's code tables. So they are suitable for applications in strongly regular graphs and secret sharing schemes.

Weighted counting of solutions to sparse systems of equations

AbstractGiven complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ > 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β > 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

Complete Weight Enumerator for a Class of Linear Codes from Defining Sets and Their Applications

Recently, linear codes over finite fields with a few weights have been extensively studied due to their applications in secret sharing schemes, authentication codes, constant composition codes. In this paper, for an odd prime p, the complete weight enumerator of a class of p-ary linear codes based on defining sets are determined. Furthermore, from the explicit complete weight enumerator of linear codes, a new class of optimal constant composition codes and several classes of asymptotically optimal systematic authentication codes are obtained.

Extremal invariant polynomials not satisfying the Riemann hypothesis

Zeta functions for linear codes were defined by Iwan Duursma in 1999 as generating functions for the Hamming weight enumerators of linear codes. They were generalized to the case of some invariant polynomials and so-called the formal weight enumerators by the present author. One of the most important problems from the applicable point of view is whether extremal weight enumerators satisfy the Riemann hypothesis. In this article, we show there exist extremal polynomials of the weight enumerator type, not being related to existing codes, which are invariant under the MacWilliams transform and do not satisfy the Riemann hypothesis. Such examples are contained in a certain invariant polynomial ring which is found by the use of the binomial moment. We determine the group which fixes the ring. To formulate the extremal property, we also establish an analog of the Mallows–Sloane bound for a certain sequence of the members in the ring.

Complete weight enumerators of some linear codes from quadratic forms

Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, a construction of q-ary linear codes with few weights employing general quadratic forms over the finite field Fq${\mathbb {F}}_{q}$ is proposed, where q is an odd prime power. This generalizes some earlier constructions of p-ary linear codes from quadratic bent functions over the prime field Fp${\mathbb {F}}_{p}$, where p is an odd prime. The complete weight enumerators of the resultant q-ary linear codes are also determined.

Macwilliams identities of linear codes over the ring F 2 + uF 2 + vF 2

The Lee weight enumerators and the complete weight enumerators for the linear codes over ring R = F2 + uF2 + vF2 are defined and Gray map φ from Rn to F23n is constructed. By proving the fact that the Gray images of the self-dual codes over R are the self-dual codes over F2, and based on the MacWilliams identities for the Hamming weight enumerators of linear codes over F2, the MacWilliams identities for Lee weight enumerators of linear codes over R are given. Further, by introducing a special variable t, the MacWilliams identities for the complete weight enumerators of linear codes over R are obtained. Finally, an example which illustrates the correctness and function of the two MacWilliams identities is provided.

MacWilliams Identities of Linear Codes over a Matrix Ring with Respect to Rosenbloom-Tsfasman Metric

 Abstract—The definitions of ρ complete weight enumerator and exact complete ρ weight enumerator over a matrix ring are given, and the MacWilliams identities with respect to RT metric for the two weight enumerators of linear codes over the matrix ring are obtained, respectively. Finally, we give an example to illustrate our obtained results.

The MacWilliams Identity of Linear Codes over M<sub>n×s</sub>(F<sub>p</sub>+uF<sub>p</sub>+vF<sub>p</sub>+uvF<sub>p</sub>) with Respect to RT Metric

The definition of the exact complete ρ weight enumerator over Mn×s(Fp+uFp+vFp+uvFp) is given, and the MacWilliams identity with respect to RT metric for the exact complete ρ weight enumerator of linear codes over Mn×s(Fp+uFp+vFp+uvFp) is obtained. Finally, a example is presented to illustrate the obtained results.

Demi-matroids from codes over finite Frobenius rings

We present a construction of demi-matroids, a generalization of matroids, from linear codes over finite Frobenius rings, as well as a Greene-type identity for rank generating functions of demi-matroids. We also prove a MacWilliams-type identity for Hamming support enumerators of linear codes over finite Frobenius rings. As a special case, these results give a combinatorial proof of the MacWilliams identity for Hamming weight enumerators of linear codes over finite Frobenius rings.

An identity between the m-spotty weight enumerators of a linear code and its dual

The m-spotty byte error control codes provide a good source for detecting and correcting errors in semiconductor memory systems using high density RAM chips with wide I/O data (e.g. 8, 16, or 32 bits). m-spotty byte error control codes are very suitable for burst correction. Here, we introduce the m-spotty weights and m-spotty weight enumerator of linear codes over the ring F_2+uF_2 and prove a MacWilliams type identity.

The MacWilliams identity for m-spotty weight enumerators of linear codes over finite fields

Some of the error control codes are applied to high speed memory systems using RAM chips with either 1-bit I/O data ( b = 1 ) or 4-bit I/O data ( b = 4 ) . However, modern large-capacity memory systems use RAM chips with 8, 16, or 32 bits of I/O data. A new class of codes called m -spotty byte error codes provides a good source for correcting/detecting errors in those memory systems that use high-density RAM chips with wide I/O data (e.g. 8, 16, or 32 bits). The MacWilliams identity provides the relation of weight distribution of a code and that of its dual code. The main purpose of this paper is to present a version of the MacWilliams identity for m -spotty weight enumerators of linear codes over arbitrary finite fields.

Code Enumerators and Tutte Polynomials

It is proved that the set of higher weight enumerators of a linear code over a finite field is equivalent to the Tutte polynomial associated to the code. An explicit expression for the Tutte polynomial is given in terms of the subcode weights. Generalizations of these results are proved and are applied to codeword m-tuples. These general results are used to prove a very general MacWilliams-type identity for linear codes that generalizes most previous extensions of the MacWilliams identity. In addition, a general and very useful matrix framework for manipulating weight and support enumerators of linear codes is presented.

The Higher Weight Enumerators of the Doubly-Even, Self-Dual $[48, 24, 12]$ Code

An efficient algorithm is presented for calculating higher weight enumerators of linear codes given generator matrices. By this algorithm, the higher weight enumerators of the unique doubly-even, self-dual code are calculated. The algorithm is based on a previously shown relationship between Tutte polynomials and higher weight enumerators.

The complete weight enumerator for codes over [formula omitted]n× s( R)

A MacWilliams identity for complete weight enumerators of codes over M n× s ( F q) endowed with a non-Hamming metric is proved in [1]. We extend the notions introduced in [1] and prove a MacWilliams identity with respect to this new metric for the complete weight enumerator of linear codes over M n× s ( R) where R is a commutative finite ring.

On Some Polynomials Related to Weight Enumerators of Linear Codes

A linear code can be thought of as a vector matroid represented by the columns of the code's generator matrix; a well-known result in this context is Greene's theorem on a connection of the weight polynomial of the code and the Tutte polynomial of the matroid. We examine this connection from the coding-theoretic viewpoint, building upon the rank polynomial of the code. This enables us to obtain bounds on all-terminal reliability of linear matroids and new proofs of two known results: Greene's theorem and a connection between the weight polynomial and the partition polynomial of the Potts model.

The Weight Enumerator of Linear Codes over GF (q^m) Having Generator Matrix over GF (q)

We have the relationships between the Hamming weight enumerator of linear codes over { GF}(q^m) which have generator matrices over { GF}(q) , the support weight enumerator and the \lambda -ply weight enumerator.

On an identity of theta functions obtained from weight enumerators of linear codes

On an identity of theta functions obtained from weight enumerators of linear codes