A serial-in-serial-out hardware architecture for systematic encoding of Hermitian codes via Gröbner bases

When a nontrivial permutation of a Hermitian code is given, the code will have a module structure over a polynomial ring of one variable. By exploiting the theory of Gröbner bases for modules, a novel and elegant systematic encoding scheme for Hermitian codes is proposed by Heegard et al. (1995). The goal of this paper is to develop a serial-in-serial-out hardware architecture, similar to a classical cyclic encoder, for such a systematic encoding scheme. Moreover, we demonstrate that under a specific permutation, the upper bounds of the numbers of memory elements and constant multipliers in the proposed architecture are both proportional to O(n), where n is the length of the Hermitian code. To encode a codeword of length n, this architecture takes n clock cycles without any latency. Therefore, the hardware complexity of the proposed architecture is much less than that of the brute-force systematic encoding by matrix multiplication.