Recently, Roth and Skachek proposed two methods for constructing nearly maximum-distance separable (MDS) expander codes. We show that through the simple modification of using mixed-alphabet codes derived from MDS codes as constituent codes in their code designs, one can obtain nearly MDS codes of significantly smaller alphabet size, albeit at the expense of a (very slight) reduction in code rate.

A q-ary (n,k)-MDS code, linear or not, satisfies n≤ q+ k−1. A code meeting this bound is said to have maximum length. Using purely combinatorial methods we show that an MDS code with n = q + k− 2 can be uniquely extended to a full length code if and only if q is even. This result is best possible in the sense that there is, for example, a non-extendable 4-ary (5,4)-MDS code. It may be that the ...

We construct a new family of quantum MDS codes from classical generalized Reed-Solomon codes and derive the necessary and sufficient condition under which these quantum codes exist. We also give code bounds and show how to construct them analytically. We find that existing quantum MDS codes can be unified under these codes in the sense that when a quantum MDS code exists, then a quantum code of...

For a linear [n, k, d] code, it is well known that d ≤ n−k+1 (see [12]), and when d = n−k+1, it is called the MDS code (maximum distance separable). For every n ≤ q + 1 there is an MDS [n, k, d] code for any given k and d satisfying d = n − k + 1. It is just the geometric codes on a rational curve (see [15]). There is a long-standing conjecture about the MDS linear codes that is called the main...

We consider the following generalization of an (n, k) MDS code for application to an erasure channel with additive noise. Like an MDS code, our code is required to be decodable from any k received symbols, in the absence of noise. In addition, we require that the noise margin for every allowable erasure pattern be as large as possible and that the code satisfy a power constraint. In this paper ...

An (n,M) vector code C ⊆ F is a collection ofM codewords where n elements (from the field F) in each of the codewords are referred to as code blocks. Assuming that F ∼= B, the code blocks are treated as l-length vectors over the base field B. Equivalently, the code is said to have the sub-packetization level l. This paper addresses the problem of constructing MDS vector codes which enable exact...

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.

Munuera, C., On MDS elliptic codes, Discrete Mathematics 117 (1993) 2799286. In this paper we give a bound for MDS (maximum distance separable) algebraic-geometric codes arising from elliptic curves. Several consequences are presented.

The parameters of a linear code C over GF(q) are given by [n, k, d], where n denotes the length, k the dimension and d the minimum distance of C . The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n−k+1. Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which hav...

An (n, k) group code over a group G is a subset of G which forms a group under componentwise group operation and can be defined in terms of n — k homomorphisms from G to G. In this correspondence, the set of homomorphisms which define Maximum Distance Separable (MDS) group codes defined over cyclic groups are characterized. Each defining homomorphism can be specified by a set of k endomorpbisms...