A theorem on the quantum evaluation of Weight Enumerators for a certain class of Cyclic Codes with a note on Cyclotomic cosets

Abstract

This note is a stripped down version of a published paper on the Potts partition function, where we concentrate solely on the linear coding aspect of our approach. It is meant as a resource for people interested in coding theory but who do not know much of the mathematics involved and how quantum computation may provide a speed up in the computation of a very important quantity in coding theory. We provide a theorem on the quantum computation of the Weight Enumerator polynomial for a restricted family of cyclic codes. The complexity of obtaining an exact evaluation is $O(k^{2s}(\log q)^{2})$, where $s$ is a parameter which determines the class of cyclic codes in question, $q$ is the characteristic of the finite field over which the code is defined, and $k$ is the dimension of the code. We also provide an overview of cyclotomic cosets and discuss applications including how they can be used to speed up the computation of the weight enumerator polynomial (which is related to the Potts partition function). We also give an algorithm which returns the coset leaders and the size of each coset from the list $\{0,1,2,...,N-1\}$, whose time complexity is soft-O(N). This algorithm uses standard techniques but we include it as a resource for students. Note that cyclotomic cosets do not improve the asymptotic complexity of the computation of weight enumerators.