Distance Enumerators for Number-Theoretic Codes

TL;DR

This paper introduces a new identity for the distance enumerator of number-theoretic codes, facilitating analysis of their error correction capabilities, and exemplifies this with the Hamming distance enumerator for VT codes.

Contribution

It establishes an identity for the distance enumerator of number-theoretic codes and derives the Hamming distance enumerator for VT codes as an example.

Findings

Derived the distance enumerator identity for number-theoretic codes

Calculated the Hamming distance enumerator for VT codes

Enhanced understanding of error correction analysis for these codes

Abstract

The number-theoretic codes are a class of codes defined by single or multiple congruences and are mainly used for correcting insertion and deletion errors. Since the number-theoretic codes are generally non-linear, the analysis method for such codes is not established enough. The distance enumerator of a code is a unary polynomial whose $i$th coefficient gives the number of the pairs of codewords with distance $i$. The distance enumerator gives the maximum likelihood decoding error probability of the code. This paper presents an identity of the distance enumerators for the number-theoretic codes. Moreover, as an example, we derive the Hamming distance enumerator for the Varshamov-Tenengolts (VT) codes.