Divisors on Division Algebras and Error Correcting Codes

Recall that a (linear) code is just a k-subspace C of k, where k is some finite field. The elements of C are referred to as codewords, and the dimension of the code, dim(C), is just the dimension of C as a k-space. The length of the code is the dimension of the ambient space k; that is, the length of C is n. The minimum distance, d(a, b) between two codewords a = (a1, . . . , an) and b = (b1, . . . , bn) is defined by d(a, b) = |{i | ai 6= bi}|, and the minimum distance of the code, d(C), is defined by d(C) = min{d(a, b) | a, b ∈ C, a 6= b}. The weight, w(a), of a codeword a = (a1, . . . , an) is defined by w(a) = |{i | ai 6= 0}|. The linearity of the code ensures that d(C) is also equal to min{w(a) | a ∈ C, a 6= 0}. ∗Supported in part by a grant from the N.S.F.