Digit-Indexed q-ary SEC-DED Codes with Near-Hamming Overhead
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917119087607808 |
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| author | Hu, Jiaxu Roche, Kenneth J. |
| author_facet | Hu, Jiaxu Roche, Kenneth J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02747 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Digit-Indexed q-ary SEC-DED Codes with Near-Hamming Overhead Hu, Jiaxu Roche, Kenneth J. Information Theory Combinatorics Number Theory 94B05 94B25 94B35 E.4; H.1.1 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. |
| title | Digit-Indexed q-ary SEC-DED Codes with Near-Hamming Overhead |
| topic | Information Theory Combinatorics Number Theory 94B05 94B25 94B35 E.4; H.1.1 |
| url | https://arxiv.org/abs/2512.02747 |