Generators and weights of polynomial codes

Abstract

Several authors have established that many classical codes are ideals in certain ring constructions. Berman, in the case of characteristic two, and Charpin, in the general case, proved that all generalized Reed-Muller codes coincide with powers of the radical of the quotient ring $A=F_q[x_1,\ldots,x_n]/(x_1^{q_1}-1,\ldots,x_n^{q_n}-1),$ where $F_q$ is a finite field, $p=\char F_q>0$ and $q_i=p^{c_i}$, for $i=1,\ldots,n$, and gave formulas for their Hamming weights. These codes form an important class containing many codes of practical value. Error-correcting codes in similar ring constructions $A$ have also been considered by Poli. Our paper contains new results that generalise and strengthen several facts obtained earlier by other authors.