Bounds on Mixed Codes with Finite Alphabets

TL;DR

This paper extends classical bounds on error-correcting codes to mixed codes over finite alphabets, providing new theoretical limits relevant for degrading storage systems.

Contribution

It generalizes key bounds like Gilbert-Varshamov and sphere-packing to mixed codes with finite alphabets, including the first Fourier-analytic proof for the non-symmetric mono-alphabetic case.

Findings

Generalized Gilbert-Varshamov, sphere-packing, Elias-Bassalygo bounds

First Fourier-analytic proof for non-symmetric mono-alphabetic case

Provides theoretical limits for mixed codes in finite alphabets

Abstract

Mixed codes, which are error-correcting codes in the Cartesian product of different-sized spaces, model degrading storage systems well. While such codes have previously been studied for their algebraic properties (e.g., existence of perfect codes) or in the case of unbounded alphabet sizes, we focus on the case of finite alphabets, and generalize the Gilbert-Varshamov, sphere-packing, Elias-Bassalygo, and first linear programming bounds to that setting. In the latter case, our proof is also the first for the non-symmetric mono-alphabetic $q$-ary case using Navon and Samorodnitsky's Fourier-analytic approach.