Concatenated Permutation Codes under Chebyshev Distance

Abstract

Permutation codes are error-correcting codes over symmetric groups. We focus on permutation codes under Chebyshev (ℓ_∞) distance. A permutation code invented by Kløve et al. is of length n, size 2^n-d and, minimum distance d. We denote the code by ϕ_n,d. This code is the largest known code of length n and minimum Chebyshev distance d > n/2 so far, to the best of the authors knowledge. They also devised efficient encoding and hard-decision decoding (HDD) algorithms that outperform the bounded distance decoding.

In this paper, we derive a tight upper bound of decoding error probability of HDD. By factor graph formalization, we derive an efficient maximum a-posterior probability decoding algorithm for ϕ_n,d. We explore concatenating permutation codes of ϕ_n,d=0 with binary outer codes for more robust error correction. A naturally induced pseudo distance over binary outer codes successfully characterizes Chebyshev distance of concatenated permutation codes. Using this distance, we upper-bound the minimum Chebyshev distance of concatenated codes. We discover how to concatenate binary linear codes to achieve the upper bound. We derive the distance distribution of concatenated permutation codes with random outer codes. We demonstrate that the sum-product decoding performance of concatenated codes with outer low-density parity-check codes outperforms conventional schemes.