ERROR DETECTION AND CORRECTION OF THE [15 8 3]-PERFECT CODE DUE TO THE AUNU AVOIDING PERMUTATION PATTERNS

Abstract

The error correction and detection of the (7 4) Hamming code and other codes of smaller lengths are demonstrated using Venn diagrams. Alon and Liu [1] about two decades ago demonstrated how a binary block code of length $\ell \geq 3$ and $n$ being the greatest integer such that $2^n-1 \leq \ell$ can be encoded and decoded using the concept of the power set. In their approach, the word length has been chosen to be the greatest integer $m$ such that $m<2^n-\binom{\tilde{n}}{0}-\binom{\tilde{n}}{1}$. They tried to generalize it on a set of $n$ elements satisfying $2^n-1 \leq \ell$. In this paper, we adopt their approach by enumerating it on the [15 8 3]perfect code constructed earlier by the authors. Our demonstration shows clearly that their approach is a suitable alternative error detection and correction scheme for this code and other binary codes satisfying these conditions.

Received: December 15, 2023;
Revised: March 28, 2024;
Accepted: April 14, 2024