Super-simple $(v,5,2)$ directed designs and their smallest defining sets with its application in LDPC codes

TL;DR

This paper proves the existence of super-simple directed designs with specific parameters for all sufficiently large v congruent to 0 or 1 mod 5, and demonstrates their application in constructing LDPC codes.

Contribution

It establishes the existence of super-simple (v,5,2) directed designs for all v ≡ 0,1 (mod 5) with v ≥ 15 and explores their use in LDPC code construction.

Findings

Existence of super-simple (v,5,2) directed designs for all v ≡ 0,1 (mod 5), v ≥ 15.

Designs with smallest defining sets containing at least half of the blocks.

Application of these designs in constructing LDPC codes.

Abstract

In this paper, we show that for all $v\equiv 0,1$ (mod 5) and $v\geq 15$, there exists a super-simple $(v,5,2)$ directed design, also for these parameters there exists a super-simple $(v,5,2)$ directed design such that its smallest defining sets contain at least half of its blocks. Also, we show that these designs are useful in constructing parity-check matrices of LDPC codes.