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

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.

 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.

As a special type of linear codes, minimal linear codes have important applications in secret sharing. Up to now, only a few infinite families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ were reported in the literature, while vast knowledge exists on the ones with $$w_{\min }/w_{\max }> 1/2$$ . Herein, $$w_{\min }$$ and $$w_{\max }$$ respectively denote the minimum and maximum nonzero Hamming weights in a linear code. Recently, several classes of linear codes with certain properties were constructed by Zhou et al. from a generic construction. The objective of this paper is to obtain four families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ from those linear codes proposed by Zhou et al. The parameters of our minimal linear codes are quite different from known ones. Based on the properties of Krawtchouk polynomials, the weight distributions of all these four families of binary linear codes are established.

Operations that combine smaller networks into a larger network in a manner such that the rate region of the larger network can be directly obtained from the associated rate regions of the smaller networks are defined and presented. The operations are selected to also have the property that the sufficiency of classes of linear network codes and the tightness of the Shannon (polymatroid) outer bound are preserved in the combination. Four such operations are defined, and the classes of linear codes considered include both scalar and vector codes. It is demonstrated that these operations enable one to obtain rate regions for networks of arbitrary size, and to also determine if some classes of linear codes are sufficient and/or the Shannon outer bound bound is tight for these networks.

Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials.

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

Two-weight or three-weight binary linear codes from cyclotomic mappings

Two-weight and three-weight linear codes based on Weil sums

Uniform constant composition codes are particular cases of constant composition codes, where every symbol of the alphabet appears λ times in the codewords. A construction of optimal uniform constant composition codes with no restrictions over the alphabet is presented.

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

A constant composition code is a special constant weight code under the restriction that each symbol should appear a given number of times in each codeword. In this paper, we give an asymptotically optimal lower bound for the maximum size of the q-ary constant composition codes with minimum distance at least 3. In addition, a number of optimum constant composition codes are obtained by using three constructions.

Determining the insdel distance of linear codes is a very challenging problem. Recently, Ji et al. [‘Strict half-Singleton bound, strict direct upper bound for linear insertion-deletion codes and optimal codes’, IEEE Trans. Inform. Theory 69 (5) (2023), 2900–2910] proposed their strict half-Singleton bound for linear codes that do not include the all-one vector. A natural question asks for linear codes with both large Hamming distance and insdel distance. Almost maximum distance separable (MDS) codes are a special kind of linear code with good Hamming error-correcting capability. We present a sufficient condition for the insdel distance of almost MDS codes to be bounded by the strict half-Singleton bound. In addition, we construct a class of two-dimensional near MDS codes that achieve the strict half-Singleton bound using twisted Reed–Solomon codes. Finally, we present a construction of optimal almost MDS insdel codes from Reed–Solomon codes for large dimensions.

On the error-correcting pair for MDS linear codes with even minimum distance

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 $F_2+vF_2$.

A classical method of constructing a linear code over $\gf(q)$ with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over $\gf(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented.