Binary Linear Codes via Zeon and Sym-Clifford Algebras

Zeon algebras have proven to be useful for enumerating structures in graphs, such as paths, trails, cycles, matchings, cliques, and independent sets. Sym-Clifford algebras have been used to enumerate walks on hypercubes without the need for adjacency matrices. In the current work, zeon (“nil-Clifford”) and “sym-Clifford” methods are used to reformulate essential concepts of binary linear coding theory. In particular, zeon and sym-Clifford methods are used to generate linear codes and to illustrate Clifford-algebraic formulations of encoding, decoding and error-correction.