Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general constr...

Given a linear code and observations from a noisy channel, the decoding problem is to determine the most likely (ML) codeword. We describe a method for approximate ML decoding of an arbitrary binary linear code, based on a linear programming (LP) relaxation that is defined by a factor graph or parity check representation of the code. The resulting LP decoder, which generalizes our previous work...

Linear codes with few weights have been an interesting subject of study for many years, as these codes have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, linear codes with a few weights are constructed from inhomogeneous quadratic functions over the finite field GF(p), where p is an odd prime. They include some earlier li...

Linear codes have been an interesting subject of study for many years. Recently, linear codes with few weights have been constructed and extensively studied. In this paper, for an odd prime p, a class of three-weight linear codes over Fp are constructed. The weight distributions of the linear codes are settled. These codes have applications in authentication codes, association schemes and data ...

Linear complementary pairs (LCP) of codes play an important role in armoring implementations against side-channel attacks and fault injection attacks. One of the most common ways to construct LCP of codes is to use Euclidean linear complementary dual (LCD) codes. In this paper, we first introduce the concept of linear codes with σ complementary dual (σ-LCD), which includes known Euclidean LCD c...

1 Lecture 1: Codes and groups 2 1.1 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Matrix representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Conway polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Linear codes: generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1...

Codes in the rank metric have been studied for the last four decades. For linear codes a Singleton-type bound can be derived for these codes. In analogy to MDS codes in the Hamming metric, we call rank-metric codes that achieve the Singleton-type bound MRD (maximum rank distance) codes. Since the works of Delsarte [3] and Gabidulin [4] we know that linear MRD codes exist for any set of paramete...

The objective of this paper is to construct a class of linear codes with two nonzero weights and three nonzero weights by using the general trace functions, which weight distributions has been determined. These linear codes contain some optimal codes, which meets certain bound on linear codes. The dual codes are also studied and proved to be optimal or almost optimal. These codes may have appli...

Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of q-ary linear codes under some certain conditions, where q is a power of a prime. The lower bound of its minimum Hamming distance is obtained. In some special cases, we evaluate the weight distributions of the linear codes by semi-p...