The Generation of $(n,n(n-1),n-1)$ Permutation Group Codes for Communication Systems

This paper concerns the construction and hardware execution of (n, n(n-1), n-1) permutation group codes (PGCs) for communication systems. Cyclic shift techniques acting on a permutation, including cyclic shift operators and their compound functions (CFs), are defined and demonstrated, then compared to other tools such as cyclic subgroups and n-dimension cyclic shift registers. A generalized finite state machine is established that can enumerate n! permutations based on complete CFs. This is a hardware-oriented algorithm that is capable of being significantly faster than all existing algorithms which we can find up to now. After this, the ideal structural features for an ordered set of n! permutations are discussed and the multiset features of (n, n(n-1), n-1)-PGCs in this ordered set are pinpointed. A simpler way of generating (n, n(n-1), n-1)-PGCs is then developed using cyclic shift techniques, where the code length n is a prime. This outperforms existing approaches using composite operation and affine transform on module-n operation. The simulation experiments in AWGN channel show that the performance on codeword error rates (WER) and signal-noise ratio (SNR) becomes better as the code length increases and the best SNR performance achieves −0.8 dB at 10
⁻⁵
WER for code length n=11.