Orthogonal Checksets in the Plane and Enumerations of the Rationals mod p.

Abstract

Binary linear codes with wordlength p ² for prime p and minimum distance at least 2s and at most 2p are constructed using as check sets all sp of the p-bit lines in G(p),a finite plane geometry mod p, which have any one of s p-distinct slopes. Minimum distances greater than 2s seem attainable for some sets of s slopes, but 2p is a firm upper bound. A norm on the rationals mod p allows choices of s p-distinct slopes which remain p’-distinct and keep minimum distance for all prime p’ > p.