Some Codes Arising from Elliptic Modular Surfaces
Tetsuji Shioda · Rikkyo University academic repository (Rikko Roots) (Rikkyo University) · 2002
For every N ≥ 3, there is a linear code over Z/NZ with some interesting properties, which arises from the Mordell-Weil lattice of the elliptic modular surface of level N . It is an [n, k, d]-code with the code length n equal to the number of cusps of the elliptic modular curve of level N and the rank k = 2 such that every code word has a constant “Bernoulli norm”. The minimum distance d is such that d/n is equal to p0/(p0 + 1) if p0 is the least prime divisor of N . Moreover it has a natural action by SL(2,Z/NZ), and a Z/NZ-valued nondegenerate bilinear pairing compatible with the action. In particular, for a prime level p ≥ 3, we obtain an [n, k, d] constant weight code over the finite field Fp = Z/pZ such that n = p2 − 1 2 , k = 2, d = p2 − p 2 which has an SL(2,Fp)-action and an invariant nondegenerate pairing.