Reversible Self-Dual Codes Over the Ring F₂ &#x002B; <italic>u</italic>F₂

Abstract

In this study, we introduce bisymmetric self-dual codes over the finite field ${\mathbb F}_{2}$ of order two. We developed a method to generate binary bisymmetric self-dual codes from a small-length bisymmetric self-dual code by increasing its length. Using this method, we produced binary bisymmetric self-dual codes and discovered that numerous such codes exhibit favorable parameters. Also, we defined the map from binary bisymmetric self-dual codes to reversible self-dual codes over the ring ${\mathbb F}_{2}+u {\mathbb F}_{2}$ . This implies that there exists a one-to-one correspondence between the bisymmetric code over ${\mathbb F}_{2}$ and the reversible self-dual code over ${\mathbb F}_{2}+u {\mathbb F}_{2}$ . Consequently, using this map on generated bisymmetric self-dual codes, we obtained reversible self-dual codes over ${\mathbb F}_{2}+u {\mathbb F}_{2}$ , which were difficult to obtain using previously known methods.

Keywords

• Code over a ring
• reversible self-dual code
• eigenvectors
• bisymmetric matrix
• bisymmetric self-dual codes