On generalized MacDonald codes

Abstract

We show that the generalized $q$-ary MacDonald codes $C_{n,u,t}\left(q\right)$ with parameters $\left[t\left[\genfrac{}{}{0ex}{}{n}{1}\right]-\left[\genfrac{}{}{0ex}{}{u}{1}\right],n,t{q}^{n-1}-q^{u-1}\right]$ are two-weight codes with nonzero weights $w_1=t{q}^{n-1}$, $w_2=t{q}^{n-1}-q^{u-1}$ and determine the complete weight enumerator of these codes. This leads to a family of strongly regular graphs with parameters $\langle q^n,q^n-q^{n-u},q^n-2q^{n-u},q^n-q^{n-u}\rangle$. Further, we show that the codes $C_{n,u,t}\left(q\right)$ satisfy the Griesmer bound and are self-orthogonal for $q=2$.