Digit-Indexed q-ary SEC-DED Codes with Near-Hamming Overhead

Abstract

We present a simple $q$-ary family of single-error-correcting, double-error-detecting (SEC--DED) linear codes whose parity checks are tied directly to the base-$p$ ($q=p$ prime) digits of the coordinate index. For blocklength $n=p^r$ the construction uses only $r+1$ parity checks -- \emph{near-Hamming} overhead -- and admits an index-based decoder that runs in a single pass with constant-time location and magnitude recovery from the syndromes. Based on the prototype, we develop two extensions: Code A1, which removes specific redundant trits to achieve higher information rate and support variable-length encoding; and Code A2, which incorporates two group-sum checks together with a 3-wise XOR linear independence condition on index subsets, yielding a ternary distance-4 (SEC--TED) variant. Furthermore, we demonstrate how the framework generalizes via $n$-wise XOR linearly independent sets to construct codes with distance $d = n + 1$, notably recovering the ternary Golay code for $n = 5$ -- showing both structural generality and a serendipitous link to optimal classical codes. Our contribution is not optimality but \emph{implementational simplicity} and an \emph{array-friendly} structure: the checks are digitwise and global sums, the mapping from syndromes to error location is explicit, and the SEC--TED upgrade is modular. We position the scheme against classical $q$-ary Hamming and SPC/product-code baselines and provide a small comparison of parity overhead, decoding work, and two-error behavior.