Triadic Codes

AbstractA new, infinite family of cyclic codes over GF(q), called triadic codes, is defined by means of idempotents. These codes are analogous to duadic codes. D-diagrams are introduced to discuss equivalence results for prime length duadic codes. The corresponding T-diagrams for triadic codes are used to prove existence theorems. In particular, triadic codes of prime lenght p over GF(q) exist iff q is a cubic residue (mod p). A new inclusion-exclusion formula for the dimensions of cyclic codes is given and used to determine possible dimensions of triadic codes. When triadic codes of prime length exist, all of these possible dimensions are attained by some triadic code. A cube-root bound is given for the minimum odd-like weight in a triadic code