Minimum Distance of Codes and Their Branching Program Complexity

The branching program is a fundamental model of (nonuniform) computation, which conveniently captures both time and space restrictions. Recently, an interesting connection between the minimum distance of a code and the branching program complexity of its encoder was established by Bazzi and Mitter. Here, we establish a relationship between the minimum distance of a linear code C and the branching program complexity of computing the syndrome function for C and/or its dual code C^perp. Specifically, let C be an (n, k, d) linear code over F_q, and suppose that there is a branching program B that computes the syndrome vector with respect to the dual code C^perp in time T and space S. We prove that the minimum distance of C is then bounded by d les 2T(S+log₂T)/klog₂q + 1. We also consider the average-case complexity in the branching program model: we show that if B computes the syndrome with respect to C^perp in expected time T and expected space S, then d les 12T(S+log₂T + 6)/klog₂q + 1. Since there are trivial branching programs that compute the syndrome vector with time-space complexity ST = O(n² log q), the bound in (2) is asymptotically tight. Furthermore, with the help of the bounds in (1) and (2), we prove the conjecture of Bazzi and Mitter that a sequence of codes whose encoder function is computable by a branching program with time-space complexity ST = o(n²) cannot be asymptotically good, for the special case of self-dual codes. Our proof of these results is based on the probabilistic method developed by Borodin-Cook and Abrahamson