On a Family of Circulant Matrices for Quasi-Cyclic Low-Density Generator Matrix Codes

TL;DR

This paper introduces a new class of sparse, invertible circulant matrices that enhance the design of quasi-cyclic low-density generator matrix codes by reducing encoding complexity without compromising code quality.

Contribution

The paper proposes a novel family of circulant matrices that are sparse, invertible, and suitable for improving quasi-cyclic LDGM code design, avoiding the drawbacks of permutation matrices.

Findings

Matrices are sparse and easily invertible.

They enable efficient encoding with fewer operations.

They do not negatively impact code minimum distance.

Abstract

We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes, that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors.