From Hamming to Hyperarcs : A Computational Tour of Algebraic Coding Theory

We develop a self-contained account of the algebraic theory of error-correcting codes over finite fields q, providing proofs or detailed proof sketches of the principal structural results: the Hamming, Singleton, Plotkin, Griesmer, and Gilbert-Varshamov bounds; the MacWilliams Identity relating the weight enumerator of a code to that of its dual; the BCH bound via Vandermonde determinants; Gleason's theorem on invariant rings of Type II self-dual codes; and Delsarte's linear programming bound via Krawtchouk polynomials. Four classical families, Hamming, Golay, BCH, and Reed-Solomon codes, are studied in detail, with explicit constructions, step-by-step decoding (syndrome tables and Berlekamp-Massey), and geometric interpretations including the Fano plane. The theoretical exposition is complemented by systematic computational investigations. We survey the gap between the best known upper and lower bounds on the minimum distance for all 8\, 128 binary code parameters with n 128, finding that 83\% of cases remain open. We verify the Main MDS Conjecture exhaustively for q with q 9, confirming the hyperoval exception for q = 4. We independently compute the unique weight enumerator of the putative 72, 36, 16 extremal Type II code and discuss the recent non-existence proof of Janusz (2022). Finally, we determine the complete weight spectra of Reed-Muller codes up to RM (2, 6) and establish the Gleason negativity threshold for extremal doubly-even self-dual codes. Applications to deep-space communication, distributed storage, and post-quantum cryptography are briefly surveyed.