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

Jiaxu Hu, Kenneth J. Roche

TL;DR

We address designing implementationally simple, non-binary SEC-DED codes with near-Hamming overhead by introducing digit-indexed parity checks tied to base-prime digits, enabling a single-pass, explicit index-based decoder. The approach is extended with A1 and A2 variants: A1 increases information rate and supports adaptive-length encoding by removing a redundant trit, while A2 introduces two group-sum checks and 3-wise XOR independence to achieve distance 4 (SEC-TED). The framework generalizes to n-wise XOR independent index sets, enabling distance d=n+1, including a ternary Golay code instance for n=4. The design emphasizes implementational simplicity and array-friendly parity structure, with comparisons to $q$-ary Hamming and SPC/product baselines and potential extensions to larger minimum distances.

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.

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

TL;DR

We address designing implementationally simple, non-binary SEC-DED codes with near-Hamming overhead by introducing digit-indexed parity checks tied to base-prime digits, enabling a single-pass, explicit index-based decoder. The approach is extended with A1 and A2 variants: A1 increases information rate and supports adaptive-length encoding by removing a redundant trit, while A2 introduces two group-sum checks and 3-wise XOR independence to achieve distance 4 (SEC-TED). The framework generalizes to n-wise XOR independent index sets, enabling distance d=n+1, including a ternary Golay code instance for n=4. The design emphasizes implementational simplicity and array-friendly parity structure, with comparisons to -ary Hamming and SPC/product baselines and potential extensions to larger minimum distances.

Abstract

We present a simple -ary family of single-error-correcting, double-error-detecting (SEC--DED) linear codes whose parity checks are tied directly to the base- ( prime) digits of the coordinate index. For blocklength the construction uses only 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 -wise XOR linearly independent sets to construct codes with distance , notably recovering the ternary Golay code for -- 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 -ary Hamming and SPC/product-code baselines and provide a small comparison of parity overhead, decoding work, and two-error behavior.

Paper Structure

This paper contains 31 sections, 4 theorems, 53 equations, 1 figure, 4 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Applications (why $q$-ary)
  3. Positioning and novelty
  4. At-a-glance comparison
  5. Contributions
  6. Prototype Ternary SEC--DED Code
  7. Definition and Description
  8. Example
  9. General Base-Prime SEC--DED Code
  10. General Description
  11. Proof
  12. Parity Check XOR Method
  13. Conception of Pair and XOR-Modulo 3
  14. Application: Index and XOR in Prototype SEC--DED Code
  15. Refine Code Designs
  16. ...and 16 more sections

Key Result

Theorem 1

Assume $x$ is a prime, given $x, y, z \in \mathbb{Z}^+$, $y, z < x$. Prove that there exists exactly one $l$ such that for $l \in \mathbb{Z}^+$ s.t. $l < x$.

Figures (1)

  • Figure 1: Example cube layout for ternary Hamming code.Blue indicate the $\cdot0$ region, red the $\cdot1$ region, and yellow the $\cdot2$ region.

Theorems & Definitions (11)

  • proof
  • Theorem 1
  • proof
  • Definition 1
  • Definition 2
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1: Weight-3 Codeword for the Ternary Prototype
  • ...and 1 more