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Function-Correcting Codes with Optimal Data Protection for Hamming Code Membership

Swaraj Sharma Durgi, Anjana A. Mahesh, Anupriya Kumari, Rajlaxmi Pandey, B. Sundar Rajan

TL;DR

It is proved that the proposed bipartite construction uniquely achieves the maximum sum-distance, the largest possible minimum distance of 2, and the minimum number of distance-2 codeword pairs, which means that the resulting max-sum SEFCCs provide significantly improved data protection and Bit Error Rate performance compared to arbitrary valid assignments.

Abstract

This paper investigates single-error-correcting function-correcting codes (SEFCCs) for the Hamming code membership function (HCMF), which indicates whether a vector in $\mathbb{F}_2^7$ belongs to the [7,4,3]-Hamming code. Necessary and sufficient conditions for valid parity assignments are established in terms of distance constraints between codewords and their nearest non-codewords. It is shown that the Hamming-distance-3 relations among Hamming codewords induce a bipartite graph, a fundamental geometric property that is exploited to develop a systematic SEFCC construction. By deriving a tight upper bound on the sum of pairwise distances, we prove that the proposed bipartite construction uniquely achieves the maximum sum-distance, the largest possible minimum distance of 2, and the minimum number of distance-2 codeword pairs. Consequently, for the HCMF SEFCC problem, sum-distance maximisation is not merely heuristic-it exactly enforces the optimal distance-spectrum properties relevant to error probability. Simulation results over AWGN channels with soft-decision decoding confirm that the resulting max-sum SEFCCs provide significantly improved data protection and Bit Error Rate (BER) performance compared to arbitrary valid assignments.

Function-Correcting Codes with Optimal Data Protection for Hamming Code Membership

TL;DR

It is proved that the proposed bipartite construction uniquely achieves the maximum sum-distance, the largest possible minimum distance of 2, and the minimum number of distance-2 codeword pairs, which means that the resulting max-sum SEFCCs provide significantly improved data protection and Bit Error Rate performance compared to arbitrary valid assignments.

Abstract

This paper investigates single-error-correcting function-correcting codes (SEFCCs) for the Hamming code membership function (HCMF), which indicates whether a vector in belongs to the [7,4,3]-Hamming code. Necessary and sufficient conditions for valid parity assignments are established in terms of distance constraints between codewords and their nearest non-codewords. It is shown that the Hamming-distance-3 relations among Hamming codewords induce a bipartite graph, a fundamental geometric property that is exploited to develop a systematic SEFCC construction. By deriving a tight upper bound on the sum of pairwise distances, we prove that the proposed bipartite construction uniquely achieves the maximum sum-distance, the largest possible minimum distance of 2, and the minimum number of distance-2 codeword pairs. Consequently, for the HCMF SEFCC problem, sum-distance maximisation is not merely heuristic-it exactly enforces the optimal distance-spectrum properties relevant to error probability. Simulation results over AWGN channels with soft-decision decoding confirm that the resulting max-sum SEFCCs provide significantly improved data protection and Bit Error Rate (BER) performance compared to arbitrary valid assignments.
Paper Structure (13 sections, 9 theorems, 5 equations, 1 figure)

This paper contains 13 sections, 9 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function-Correcting Codes
  4. $[7,4,3]$-Hamming Code
  5. Problem Setup
  6. Main Results
  7. Parity Assignment and FCC Validity
  8. Distance Optimization Among Valid FCCs
  9. Optimal SEFCC Construction
  10. FCC Construction Using Bipartition
  11. Distance Properties of the Constructed FCCs
  12. Simulation Results
  13. Conclusion

Key Result

Theorem 1

For any binary-valued function $f:\mathbb{F}_2^k \rightarrow \mathbb{F}_2$, define $f^{-1}(0) \triangleq \{\mathbf{u} \in \mathbb{F}_2^k \mid f(\mathbf{u})=0\}$ and $f^{-1}(1) \triangleq \{\mathbf{u} \in \mathbb{F}_2^k \mid f(\mathbf{u})=1\}$. An optimal FCC $\mathcal{C}$ that assigns the same pari

Figures (1)

  • Figure 1: BER and FER comparison of $\mathcal{C}_1$ and $\mathcal{C}_2$.

Theorems & Definitions (14)

  • Definition 1: (Systematic) Function-Correcting Code
  • Definition 2: Optimal redundancy
  • Definition 3: Hamming Distance matrix
  • Definition 4: Sum-Distance
  • Theorem 1: Optimal FER FCC
  • Proposition 1: Complementary Parity Rule
  • Proposition 2: Codeword-Neighbour Parity Rule
  • Theorem 2: Valid SEFCC
  • Theorem 3: Condition for $d_{\min}=2$
  • Lemma 1
  • ...and 4 more