Optimal design, decoding, and minimum distance analysis of Goppa codes using heuristic method

Abstract

Error-correcting codes are crucial to ensure data reliability in communication systems often affected by transmission noise. Building on previous successful applications of our heuristic method degenerate quantum simulated annealing (DQSA) to Bose–Chaudhuri–Hocquenghem (BCH) and quadratic residue (QR) codes. This paper proposes two algorithms designed to address two coding problems for Goppa codes. DQSA-dmin computes the minimum distance (dmin) while DQSA-Dec, serves as a hard decoder optimized for additive white gaussian noise (AWGN) channels. We validate DQSA-dmin comparing its computed minimum distances with theoretical estimates for algebraically constructed Goppa codes, showing accuracy and efficiency. DQSA-dmin further used to find the optimal Goppa codes that reach the lower bound of dmin for linear codes known in the literature and stored in Marcus Grassl's online database. Indeed, we discovered 12 Goppa codes reaching this lower bound. For DQSA-Dec, experimental results show that it obtains a bit error rate (BER) of 10-5 when SNR=7.5 for codes with lengths less than 65, which is very interesting for a hard decoder. Additionally, a comparison with the Paterson algebraic decoder specific to this code family shows that DQSA-Dec outperforms it with a 0.6 dB coding gain at BER=10-4. These findings highlight the effectiveness of DQSA-based algorithms in designing and decoding Goppa codes.