Weight Enumerators and Gray Maps of Linear Codes over Rings

The MacWilliams identity for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" display="inline" overflow="scroll"><mml:mi>m</mml:mi></mml:math>-spotty weight enumerators of linear codes over finite fields

 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.

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.

This paper examines the w w -weight enumerators of weights w w with maximal symmetry over finite chain rings and matrix rings over finite fields. In many cases, including the homogeneous weight, the MacWilliams identities for w w -weight enumerators fail because there exist two linear codes with the same w w -weight enumerator whose dual codes have different w w -weight enumerators.

A partial description of (complete) weight enumerators of linear self-dual codes is given. 0. Let F = Z=pZ, where p is a prime number. If C is a linear code on F of length n, i.e., a linear subspace in F n , then its (complete) weight enumerator WC is deÞned to be X u2C e Y a2F x s a(u) a e

Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are still under investigation. This work covers some of the developments regarding the weight enumerator and the capacity-achieving properties of RM codes, as well as some of the algorithmic developments. In particular, it discusses connections established between RM codes, thresholds of Boolean functions, polarization theory, hypercontractivity, and the techniques of approximating low weight codewords using lower degree polynomials (when codewords are viewed as evaluation vectors of degree r polynomials in m variables). It then overviews some of the algorithms for decoding RM codes, giving both algorithms with provable performance guarantees for every block length, as well as algorithms with state-of-the-art performances in practical regimes, which do not perform as well for large block length. Finally, some applications of RM codes in theoretical computer science and signal processing are given.

Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are still under investigation. This paper covers some of the recent developments regarding the weight enumerator and the capacity-achieving properties of RM codes, as well as some of the algorithmic developments. In particular, the paper discusses the recent connections established between RM codes, thresholds of Boolean functions, polarization theory, hypercontractivity, and the techniques of approximating low weight codewords using lower degree polynomials (when codewords are viewed as evaluation vectors of degree r polynomials in m variables). It then overviews some of the algorithms for decoding RM codes. It covers both algorithms with provable performance guarantees for every block length, as well as algorithms with state-of-the-art performances in practical regimes, which do not perform as well for large block length. Finally, the paper concludes with a few open problems.

New metrics and distances for linear codes over the ring Fq[u]/(u) are defined, which generalize the Gray map, Lee weight, and Bachoc weight; and new bounds on distances are given. Two characterizations of self-dual codes over Fq[u]/(u) are determined in terms of linear codes over Fq. An algorithm to produce such self-dual codes is also established. 1.Introduction. Many optimal codes have been obtained by studying codes over general rings rather than fields. Lately, codes over finite chain rings (of which Fq[u]/(u) is an example) have been a source of many interesting properties [6, 8, 3]. Gulliver and Harada [4] found good examples of ternary codes over F3 using a particular type of Gray map. Siap and Ray-Chaudhuri in [9] established a relation between codes over Fq[u]/(u − a) and codes over Fq, which was used to obtained new codes over F3 and F5. In this paper we present a certain generalization of the method used in [4] and [9], defining a family of metrics for linear codes over Fq[u]/(u) and obtaining as particular examples the Gray map, the Gray weight, the Lee weight and the Bachoc weight. For the latter, we give a new bound on the distance of those codes. It also shows that the Gray images of codes over F2 + uF2 are more powerful than codes obtained by the so-called u—u+v condition. With these tools in hand, we study conditions for self-duality of codes over Fq[u]/(u). In [7] the authors study the case of self-dual cyclic codes in terms of the generator polynomials. In this paper we study self-dual codes in terms of linear codes over Fq that are obtained as images under the maps defined on the first part of the paper. We provide a way to construct many self-dual codes over Fq starting from a self-dual code over Fq[u]/(u). We also study self-dual codes in terms of the torsion codes, and provide a way to construct many self-dual codes over Fq[u]/(u) starting from a self-orthogonal code over Fq. Our results contain many of the properties studied by Bachoc for self-dual codes over F3 + uF3 in [1]. 2. Metric for Codes over Fq[u]/(u). We will use R(q, t) to denote the commutative ring Fq[u]/(u). The q elements of this ring can be represented in two different forms, and we will use the most appropriate in each case. First, we can use the ∗The project was partially supported by Office of Research of the University of Michigan-Flint. †2000 Mathematics Subject Classification: Primary: 94B05, 94B60, Secondary: 11T71. ‡

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4^m ≥ 16. At length 16 they coincide to give the Nordstrom-Robinson code. This paper constructs a nonlinear (64, 2^37, 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers modulo 4 using Galois rings. The Nordstrom-Robinson code is defined in this same way, and like the Nordstrom-Robinson code, the new code is better than any linear code that is presently known.

Linear codes constitute an important family of error-correcting codes and have a rich algebraic structure. Initially, these codes were studied with respect to the Hamming metric; while for the past few years, they are also studied with respect to a non-Hamming metric, known as the Rosenbloom–Tsfasman metric (also known as RT metric or ρ metric). In this paper, we introduce and study the split ρ weight enumerator of a linear code in the R-module Mn×s(R) of all n × s matrices over R, where R is a finite Frobenius commutative ring with unity. We also define the Lee complete ρ weight enumerator of a linear code in Mn×s(ℤk), where ℤk is the ring of integers modulo k ≥ 2. We also derive the MacWilliams identities for each of these ρ weight enumerators.

It is shown that a finite ring R is a Frobenius ring if and only if $_R(R/\hbox {Rad}\, R)\cong \hbox {Soc}\, (_RR)$ . Other combinatorial characterizations of finite Frobenius rings are presented which have applications in the theory of linear codes over finite rings.

It is very well known that algebraic structures have valuable applications in the theory of error-correcting codes. Blake [Codes over certain rings, Inform. and Control 20 (1972) 396–404] has constructed cyclic codes over ℤm and in [Codes over integer residue rings, Inform. and Control 29 (1975), 295–300] derived parity check-matrices for these codes. In [Linear codes over finite rings, Tend. Math. Appl. Comput. 6(2) (2005) 207–217]. Andrade and Palazzo present a construction technique of cyclic, BCH, alternant, Goppa and Srivastava codes over a local finite ring B. However, in [Encoding through generalized polynomial codes, Comput. Appl. Math. 30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text] and encoding, Comput. Math. Appl. 62 (2011) 1645–1654], Shah et al. extend this technique of constructing linear codes over a finite local ring B via monoid rings [Formula: see text], where p = 2 and k = 1, 2, respectively, instead of the polynomial ring B[X]. In this paper, we construct these codes through the monoid ring [Formula: see text], where p = 2 and k = 1, 2, 3. Moreover, we also strengthen and generalize the results of [Encoding through generalized polynomial codes, Comput. Appl. Math.30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text]] and [Encoding, Comput. Math. Appl.62 (2011) 1645–1654] to the case of k = 3.

In this paper, we study a type of mixed alphabet codes, called [Formula: see text]-linear codes, where [Formula: see text] with [Formula: see text]. First, we define and discuss several weight enumerators of [Formula: see text]-linear codes, including Hamming weight, Lee weight and symmetrized weight enumerators. MacWilliams identity connects the weight enumerator of codes and their dual codes. Further, we present the MacWilliams identities about Lee weight and complete weight over the ring [Formula: see text]. Finally, several examples are presented to illustrate our main results.

We begin with general properties of minimal supports. Then we evaluate the parameters of the distribution of the number of minimal supports in random linear codes, s tudy their asymptot ic behaviour, and discuss its consequences for some max imum likelihood decoding algorithms. In the second part , we construct explicitly sets of minimal supports in Hamming codes and second order Reed-Muller codes. Next we propose a generalization of minimal supports for codes over finite commutat ive rings and evaluate them for the quaternary Kerdock code and quaternary Hamming code. One application of minimal supports is in cryptography, namely, in the socalled linear secret-sharing schemes [1]. We briefly outline this connection and present an analog of the main result of [1] for codes over rings. Since minimal supports in linear codes correspond to cycles in representable matroids, our results have a clear interpretation in the frame of matroid theory. Details and proofs can be found in [2, 3].

Modular invariance property of association schemes is recalled in connection with our joint work with François Jaeger. Then we survey codes over F 2 discussing how codes, through their (various kinds of) weight enumerators, are related to (various kinds of) modular forms through polynomial invariants of certain finite group actions and theta series. Recently, not only codes over an arbitrary finite field but also codes over finite rings and finite abelian groups are considered and have been studied extensively. We show how the determination of the solutions of the modular invariance property of finite abelian groups (our joint work with Jaeger) is used to define the concept of Type II codes over arbitrary finite abelian groups. As an example of the usefulness of such Type II codes, we give an application to hermitian modular forms.