On generic erasure correcting sets and related problems

TL;DR

This paper advances the understanding of generic erasure correcting sets by establishing improved bounds, exploring their relation to s-wise intersecting codes, and applying hypergraph covering techniques for better upper bounds.

Contribution

It provides improved bounds on the minimum size of generic erasure correcting sets and their stronger variants, and connects these concepts to s-wise intersecting codes and hypergraph covering methods.

Findings

Derived better lower and upper bounds for F(r,s)

Improved bounds for G(r,s), the size of (r,s)-sets

Connected erasure correcting sets to s-wise intersecting codes and hypergraph covering

Abstract

Motivated by iterative decoding techniques for the binary erasure channel Hollmann and Tolhuizen introduced and studied the notion of generic erasure correcting sets for linear codes. A generic $(r,s)$--erasure correcting set generates for all codes of codimension $r$ a parity check matrix that allows iterative decoding of all correctable erasure patterns of size $s$ or less. The problem is to derive bounds on the minimum size $F(r,s)$ of generic erasure correcting sets and to find constructions for such sets. In this paper we continue the study of these sets. We derive better lower and upper bounds. Hollmann and Tolhuizen also introduced the stronger notion of $(r,s)$--sets and derived bounds for their minimum size $G(r,s)$. Here also we improve these bounds. We observe that these two conceps are closely related to so called $s$--wise intersecting codes, an area, in which $G(r,s)$ has been studied primarily with respect to ratewise performance. We derive connections. Finally, we observed that hypergraph covering can be used for both problems to derive good upper bounds.