Systematic Codes Correcting Two Deletions with Three Redundant Symbols over Large Alphabets

Abstract

A literal-systematic $q$-ary code leaves its $n$ information symbols unchanged and appends deletion-prone checks. For $n\ge2$, a known lower bound requires three such checks to correct two arbitrary deletions. We give a uniform construction meeting this minimum over a power-of-two alphabet $q<2^{32}n^{10}$, with polynomial-time encoding and decoding. It combines boundary authentication, symmetric field checks, and a pointwise computable coloring of a sparse conflict graph. For one adjacent burst of length at most two, we give a closed-form construction with three check symbols over $q_{\mathrm{adj}}<(n+2)^4$. Alphabet size and fixed-alphabet bit redundancy remain separate objectives.