Iterative methods to build LG-matrices and Applications

TL;DR

This paper introduces a recursive algorithm for constructing special classes of (0,1)-matrices, explores their properties, and applies them to develop new isodual linear codes with analyzed weight distributions and bounds.

Contribution

It presents a novel recursive method for building (0,1)-matrices and applies this to construct and analyze new isodual linear codes with specific properties.

Findings

Constructed two families of (0,1)-matrices with the algorithm.

Developed new isodual linear codes with weight distribution analysis.

Provided bounds for the minimum distance of these codes.

Abstract

In this paper we give a recursive algorithm to construct two families of $(0,1)$-matrices, one sparse regular and the other dense. We study various properties of the two families of $(0,1)$-matrices built with our algorithm. We present a new construction of two clases of isodual linear codes, one is the low density generator matrix codes and other is the dense linear codes, for both codes we obtain the polynomial of the distribution of weights, a bound for the minimum distance and we apply to these codes the efficient encoders based on approximate lower triangulations developed by Richardson-Urbanke. We identify the unique $(0,1)$-matrices, up basis change, associated with the geometry of the Lagrangian-Grassmannian variety.