Near MDS poset codes and distributions

We study $q$-ary codes with distance defined by a partial order of the coordinates of the codewords. Maximum Distance Separable (MDS) codes in the poset metric have been studied in a number of earlier works. We consider codes that are close to MDS codes by the value of their minimum distance. For such codes, we determine their weight distribution, and in the particular case of the "ordered metric" characterize distributions of points in the unit cube defined by the codes. We also give some constructions of codes in the ordered Hamming space.Comment: 13 pages, 1 figur

Codes with a poset metric

AbstractNiederreiter generalized the following classical problem of coding theory: given a finite field Fq and integers n > k ⩾ 1, find the largest minimum distance achievable by a linear code over Fq of length n and dimension k. In this paper we place this problem in the more general setting of a partially ordered set and define what we call poset-codes. In this context, Niederreiter's setting may be viewed as the disjoint union of chains. We extend some of Niederreiter's bounds and also obtain bounds for posets which are the product of two chains

Combinatorial channels from partially ordered sets

A central task of coding theory is the design of schemes to reliably transmit data though space, via communication systems, or through time, via storage systems. Our goal is to identify and exploit structural properties common to a wide variety of coding problems, classical and modern, using the framework of partially ordered sets. We represent adversarial error models as combinatorial channels, form combinatorial channels from posets, identify a structural property of posets that leads to families of channels with the same codes, and bound the size of codes by optimizing over a family of equivalent channels. A large number of previously studied coding problems that fit into this framework. This leads to a new upper bound on the size of s-deletion correcting codes. We use a linear programming framework to obtain sphere-packing upper bounds when there is little underlying symmetry in the coding problem. Finally, we introduce and investigate a strong notion of poset homomorphism: locally bijective cover preserving maps. We look for maps of this type to and from the subsequence partial order on q-ary strings