How Many Weights Can a Quasi-Cyclic Code Have?

We investigate the largest number of nonzero weights of quasi-cyclic codes. In particular, we focus on the function $\Gamma _{Q}(n,\ell,k,q)$ , that is defined to be the largest number of nonzero weights a quasi-cyclic code of index $\gcd (\ell,n)$ , length $n$ and dimension $k$ over $\mathbb F_{q}$ can have, and connect it to similar functions related to linear and cyclic codes. We provide several upper and lower bounds on this function, using different techniques and studying its asymptotic behavior. Moreover, we determine the smallest index for which a $q$ -ary Reed-Muller code is quasi-cyclic, a result of independent interest. [ABSTRACT FROM AUTHOR]