Hsiao-Code Check Matrices and Recursively Balanced Matrices

TL;DR

This paper discusses the construction of check matrices for Hsiao codes, introduces an improved algorithm, and corrects previous analysis errors, enhancing the practical generation of these matrices.

Contribution

It presents a modified, more efficient algorithm for generating Hsiao code check matrices and corrects earlier analysis errors, facilitating better practical implementation.

Findings

Algorithm attains optimal average-case performance with divide-and-conquer.

Modified algorithm improves efficiency and effectiveness.

Corrected previous analysis errors in the algorithm.

Abstract

The key step of generating the well-known Hsiao code is to construct a {0,1}-check-matrix in which each column contains the same odd-number of 1's and each row contains the same number of 1's or differs at most by one for the number of 1's. We also require that no two columns are identical in the matrix. The author solved this problem in 1986 by introducing a type of recursively balanced matrices. However, since the paper was published in Chinese, the solution for such an important problem was not known by international researchers in coding theory. In this note, we focus on how to practically generate the check matrix of Hsiao codes. We have modified the original algorithm to be more efficient and effective. We have also corrected an error in algorithm analysis presented in the earlier paper. The result shows that the algorithm attained optimum in average cases if a divide-and-conquer technique must be involved in the algorithm.