Binary Cyclic codes with two primitive nonzeros

Abstract

In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of $\Tr(x^d)$ over $\F_{2^{m}}$ in the case where $m=2t$, $d=3+2^{t+1}$ and $\gcd (d,2^m-1)=1$.

11 pages, submitted to the special volumn of Science Sinica Mathematics on Coding, Cryptography and Combinatorics