Abstract
This paper focuses on correcting a theorem relating to the construction of permutation group codes (PGCs). The theorem in question assumed that affine transformation could be used to enumerate all the code words of a permutation array with a minimum Hamming distance (MHD) of n - 1 for any n > 1. This assumption was founded upon the proposition that, if the code length, n, is a prime power, then the maximum cardinality of the code will be n(n - 1) and its MHD will be n - 1. However, two typical algebraic methods, one relying on affine transformation and another upon the composite operation of two small subgroups of a symmetric group, can violate this proposition. This is because it is only when n is a prime rather than a prime power that they can enumerate all the code words of an (n; n (n - 1) ; n - 1)-PGC. By investigating how the range of n impacts upon the cardinality and MHD of a code, we provide a corrective theorem that stipulates the construction of (p q ; p 2q (1 - 1/p); p q (1 - 1/p))-PGCs when n = pq is a prime power, wherep is a prime and q > 1. On the basis of this theorem, and under the condition of n being the power of 2, we construct a (2 q ; 2 2q-1 ; 2 q-1 )-PGC and present an encoder that can map a k-bit binary information sequence to an n = 2 2q -dimension permutation code word. The natural array structure of (2 q ; 2 2q-1 ; 2 q-1 )-PGCs makes them especially well-suited to forming the basis of low-complexity encoders. We present simulation-based experiments that show that, as the code length increases, the performance of these codes improves. The best performance is for a (16; 128; 8)-PGC, which can achieve -3.8 dB with a word error rate of 10 -7 .
Highlights
This paper focuses on correcting an important error in existing approaches to the construction of permutation group codes (PGC)
The remainder of this paper is organized as follows: In Section II, we show that the permutation set based on affine transformations is a PGC and, from the perspective of affine transformations, examine why the range of n > 1 can affect the parameters μ and d in two small subgroups, i.e. the maximum one-fixed-point subgroup, Ln, about the single point, n ∈ Zn, and the translation subgroup, Gn
3) It needs to be emphasized that we have not proved that the Proposition and Theorem 3 in [2] are untrue for the permutation codes generated by any other methods beyond the algebraic methods presented here, e.g., the use of a random search method in Sn
Summary
This paper focuses on correcting an important error in existing approaches to the construction of permutation group codes (PGC). 20 and 21] that the structures of all sharply k-transitive groups are known for k ≥ 2 It took until 2019 and Theorem 5 in [9] for it to be possible to clearly summarize four methods by which explicit algebraic expressions and/or basic unit circuits could be used to generate all the code words of an (n; n(n − 1); n − 1)-PGC under the condition of n being a prime number. These methods are: composition; affine transformation; the compound function of cyclic shift operators; and n-SR. We will determine and discuss the range of the parameters of this PGC, because the error in Ding et al.’s Theorem 2 [2] relates to the way in which the range of the code length, n, impacts on the code parameters, μ and d
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