Spectrum of Sizes for Perfect Deletion-Correcting Codes

TL;DR

This paper characterizes all possible sizes of perfect deletion-correcting codes for specific lengths and alphabet sizes, revealing the spectrum of code sizes and their variations.

Contribution

It completely determines the spectrum of sizes for perfect 1-deletion-correcting codes of length three and nearly all for 2-deletion-correcting codes of length four.

Findings

Spectrum of sizes for perfect 1-deletion codes of length three is fully characterized.

Almost all sizes for perfect 2-deletion codes of length four are determined.

Remaining cases for 2-deletion codes are identified as open problems.

Abstract

One peculiarity with deletion-correcting codes is that perfect $t$-deletion-correcting codes of the same length over the same alphabet can have different numbers of codewords, because the balls of radius $t$ with respect to the Levenshte\u{\i}n distance may be of different sizes. There is interest, therefore, in determining all possible sizes of a perfect $t$-deletion-correcting code, given the length $n$ and the alphabet size~$q$. In this paper, we determine completely the spectrum of possible sizes for perfect $q$-ary 1-deletion-correcting codes of length three for all $q$, and perfect $q$-ary 2-deletion-correcting codes of length four for almost all $q$, leaving only a small finite number of cases in doubt.