- Published 2006 in 2006 IEEE International Symposium on Information…

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<sup>perp</sup>. Specifically, let C be an (n, k, d) linear code over F<sub>q</sub>, and suppose that there is a branching program B that computes the syndrome vector with respect to the dual code C<sup>perp</sup> in time T and space S. We prove that the minimum distance of C is then bounded by d les 2T(S+log<sub>2</sub>T)/klog<sub>2</sub>q + 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<sup>perp</sup> in expected time T and expected space S, then d les 12T(S+log<sub>2</sub>T + 6)/klog<sub>2</sub>q + 1. Since there are trivial branching programs that compute the syndrome vector with time-space complexity ST = O(n<sup>2</sup> 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(n<sup>2</sup>) 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

@article{Santhi2006MinimumDO,
title={Minimum Distance of Codes and Their Branching Program Complexity},
author={N. Santhi and A. Vardy},
journal={2006 IEEE International Symposium on Information Theory},
year={2006},
pages={1490-1494}
}