A Note on Weight Enumerators of Linear Self-Dual Codes

The main focus in this thesis is linear codes over rings. In the first part, we look at linear codes over Galois rings, and using the homogeneous weight, we improve upon Wilson's results about the prime power that divides the coefficients of the homogeneous weight enumerators of these codes. We also prove that our results are best possible. Our results about homogeneous weight enumerators of linear codes over Galois rings generalize the results that we have for the Lee weight enumerators of linear codes over the ring of integers modulo 4. We also consider other weight enumerators, in particular the complete weight enumerators of linear codes and we obtain MacWilliams-like identities for these weight enumerators considering different rings. These MacWilliams-like identities lead to MacWilliams identities for the Hamming weight enumerators of linear codes over rings. We also give a counter-example to show that we cannot have MacWilliams-like identities for the Euclidean weight enumerators of linear codes over the ring of integers modulo 4. We also look at Gray maps from Galois rings to the finite field of prime size. We first give an application of the distance preserving Gray maps to obtain good ternary codes, and then we give an inductive algebraic construction of a distance preserving Gray map from Galois rings with the homogeneous distance to the field of prime size with the Hamming distance as well as a combinatorial construction. By using the Gray maps, we obtain some results about the weight enumerators of linear codes over the ring of dual numbers. In the last part of this work, we consider the permutation invariance of binary codes and the connection with linearity over certain rings. Among the rings considered, we have different variations and extensions of the ring of dual numbers. In this context we consider the Reed-Muller codes and answer questions about permutation invariance of Reed-Muller codes under certain permutation groups and the linearity of Reed-Muller codes over these aforementioned rings.

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.

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.

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.

In this chapter, we will explain the construction of linear block codes. Since linear block codes are vector subspaces, we advise to the reader to study Chap. 1 before proceeding to Chap. 2. Without the knowledge of basic concepts of linear algebra, such as groups, fields, vector spaces, and subspaces, it is difficult to comprehend the philosophy behind the concept of linear block codes. We advise the reader to study the section related to the philosophy of channel encoding before studying the sections related to the encoding and decoding operations. We explain the purpose of channel encoding in Sect. 2.5. After introducing the construction of linear block codes, we explain the encoding operation and give some fundamental definitions such as minimum distance, generator matrix, and parity check matrix. In sequel, we explain equivalent codes, error detection and correction capability of linear block codes, and Hamming spheres.

Near MDS codes of non-elliptic-curve type from Reed-Solomon codes

Nonlinear codes defined by quadratic forms over GF(2)

This study defined the ring $F_2+vF_2$, as well as its complete weight enumerator and symmetric weight enumerator. By introducing a special variable $t$, we deduced two Macwilliams identities for the two weight enumerators of the linear code and the dual code over the ring $F_2+vF_2$.

 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.

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted by Chinen in connection with zeta functions and Riemann hypothesis for linear codes. In this paper, we establish a relation between formal weight enumerators and Chebyshev polynomials. Specifically, the condition for the existence of formal weight enumerators with prescribed parameters $$(n,\varepsilon ,q)$$ is given in terms of Chebyshev polynomials. According to the parity of n and the sign $$\varepsilon$$ , the four kinds of Chebyshev polynomials appear in the statement of the result. Further, we obtain explicit expressions of formal weight enumerators in the case where n is odd or $$\varepsilon =-1$$ using Dickson polynomials, which generalize Chebyshev polynomials. We also state a conjecture from a viewpoint of binomial moments, and see that the results in this paper partially support the conjecture.

Recently the notion on binary codes called Z/sub 4/-linearity was introduced. This notion explains why Kerdock codes and Delsarte-Goethals codes admit formal duals in spite of their nonlinearity. The "Z/sub 4/-duals" of these codes (called "Preparata" and "Goethals" codes) are new nonlinear codes which admit simpler decoding algorithms than the previously known formal duals (the generalized Preparata and Goethals codes). We prove, by using the notion of exact weight enumerator, that the relationship between any Z/sub 4/-linear code and its Z/sub 4/-dual is stronger than the standard formal duality and we deduce the weight enumerators of related generalized codes.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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.

This thesis consists of two independent chapters. The first chapter concerns the Smith Normal Form (SNF) over the integers ℤ of integral matrices. We consider the SNF of a matrix A to be the ratio of two ℤ-modules -- a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results. The new results concern the relationship between the eigenvalues of an integral matrix and its SNF. In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group. Bounds are also given for the exponent of the Smith group. In some cases, these are best possible. The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices. The latter results are extended to rectangular matrices. Numerous examples are given throughout, along with many conjectures based on computation. The second chapter generalizes the work of Pless, et al. on duadic codes and Q-codes. We take abelian group codes to be ideals in the group ring F[G], where G is a finite abelian group of odd order n and F is a finite field with characteristic relatively prime to n. We define generalized Q-codes from a pair of idempotents of F[G] and an automorphism of G which together obey two simple equations. These codes are (n, (n+1)/2) and (n, (n-1)/2) linear codes. We show that all of the properties of duadic and Q-codes generalize. In particular, we extend the results on the relationship of these codes to projective planes with regular automorphism group G. When F has characteristic 2, we give simple numerical conditions on G and F which determine when generalized Q-codes exist. We also give some techniques for constructing these codes.

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. ‡

We introduce codes over the ring **. We relate self-dual codes over this ring to quaternionic unimodular lattices and to self-dual codes over ** via a gray map. We study a connection between the complete weight enumerators of codes over the quaternionic ring Σ2m and Jacobi forms over the half-space of quaternions. This motivates us to construct an algebra homomorphism from a certain invariant polynomial ring, where the complete weight enumerators belong, to the ring of Jacobi forms over the quaternions. Higher genus modular forms over the quaternions are also constructed using joint weight enumerators of codes.