On constructions and parameters of symmetric configurations v_{k}

TL;DR

This paper explores the parameters of symmetric configurations, introduces new construction methods, and extends the range of parameters for applications like LDPC codes, with a focus on cyclic configurations and their mathematical tools.

Contribution

It surveys existing constructions, proposes new methods, and derives new bounds for symmetric configurations, especially cyclic ones, enhancing their applicability in coding theory.

Findings

New parameters for symmetric configurations, especially cyclic ones.

New upper bounds on the minimal integer E(k) for configurations.

Extended parameter ranges for LDPC and related codes.

Abstract

The spectrum of possible parameters of symmetric configurations is investigated. We both survey known constructions and results, and propose some new construction methods. Many new parameters are obtained, in particular for cyclic symmetric configurations, which are equivalent to deficient cyclic difference sets. Both Golomb rulers and modular Golomb rulers are a key tool in our investigation. Several new upper bounds on the minimum integer E(k) such that for each v >= E(k) there exists a symmetric configuration v_{k} are obtained. Upper bounds of the same type are provided for cyclic symmetric configurations. From the standpoint of applications, it should be noted that our results extend the range of possible parameters of LDPC codes, generalized LDPC codes, and quasi-cyclic LDPC codes.