This paper presents lower and upper bounds on the number of parity-check digits required for a linear code that corrects random errors and errors which are in the form of low-density bursts.

This article reviews some of the principal and recently-discovered lower and upper bounds on the maximum size of (n, r)-arcs in PG(2, q), sets of n points with at most r points on a line. Some of the upper bounds are used to improve the Griesmer bound for linear codes in certain cases. Also, a table is included showing the current best upper and lower bounds for q ≤ 19, and a number of open pro...

The paper gives a matrix-free presentation of the correspondence between full-length linear codes and projective multisets. It generalizes the BrouwerVan Eupen construction that transforms projective codes into two-weight codes. Short proofs of known theorems are obtained. A new notion of self-duality in coding theory is explored. 94B05, 94B27, 51E22.

In this work, we consider a classification of infinite families of linear codes which achieve the Griesmer bound, using the projective dual transform. We investigate the correspondence between families of linear codes with given properties via dual transform.

The MacWilliams Extension Theorem states that each linear isometry of a linear code extends to a monomial map. Unlike the linear codes, in general, additive codes do not have the extension property. In this paper, an analogue of the extension theorem for additive codes in the case of additive MDS codes is proved. More precisely, it is shown that for almost all additive MDS codes their additive ...

Binary MDS linear codes are well-known. We prove that any binary MDS systematic non-linear code is equivalent to a linear code and hence we classify them completely.

We study formulae to count the number of binary vectors of length n that are linearly independent k at a time where n and k are given positive integers with 1 ≤ k ≤ n. Applications are given to the design of hypercubes and orthogonal arrays, pseudo (t, m, s)-nets and linear codes. 1991 AMS(MOS) Classification: Primary: 11T30; secondary: 05B15.

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by q(r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on q(r, 3). General constructions are given and upper bounds on q(r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.

A linear code of length n and dimension k is called a [n, k]-code. If G is a r × n matrix over GF (q), whose rows form a basis for C, then G is a generator matrix for C. As with every matrix, there is an associated matroid, which in this case depends only on C. Hence the matroid is M(C). The dual code, C∗, of C, is defined by C∗ = {v ∈ GF (q)n|v · w = 0 ∀w ∈ C}. If C is a [n, k]-code, then C∗ i...