Weight Spectrum of Quasi-Perfect Binary Codes with Distance 4

TL;DR

This paper analyzes the weight spectrum of quasi-perfect binary codes with distance 4, providing recursive formulas and applications to error correction probability estimates.

Contribution

It introduces exact recursive formulas for the weight spectrum of quasi-perfect codes and their duals, advancing understanding of their structure and error correction capabilities.

Findings

Recursive formulas for weight spectrum derived

Lower bounds for erasure correction probability established

Application to codes like extended Hamming and Panchenko demonstrated

Abstract

We consider the weight spectrum of a class of quasi-perfect binary linear codes with code distance 4. For example, extended Hamming code and Panchenko code are the known members of this class. Also, it is known that in many cases Panchenko code has the minimal number of weight 4 codewords. We give exact recursive formulas for the weight spectrum of quasi-perfect codes and their dual codes. As an example of application of the weight spectrum we derive a lower estimate for the conditional probability of correction of erasure patterns of high weights (equal to or greater than code distance).