Single-Deletion Recovery in Linear Codes: A Dual-Filter Criterion and Algebraic Decoding Algorithm
Abstract
Deletion errors pose fundamentally different challenges from substitution errors: when a symbol is removed, all subsequent positions shift, destroying synchronization and making classical syndrome decoding inapplicable. This paper proposes an algebraic algorithm that recovers a single deleted symbol using only the parity-check matrix of any existing linear code over GF(q), computing the missing symbol by field inversion rather than exhaustive search. A necessary and sufficient condition for unambiguous recovery is proven: two codewords conflict if and only if both a weighted-sum filter and a parity filter fail simultaneously. An offline H-matrix screening algorithm identifies all conflict-free subsets for any parity-check matrix without encoder modification.
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