Weight spectra of some families of GRM codes

Abstract

Let ${\mathrm{RM}}_q(r,m)$ denote the generalized Reed--Muller code of order $r$ and length $q^m$ over the finite field ${\mathbb{F}}_q$. We completely determine the weight spectra of ${\mathrm{RM}}_3(2m-i,m)$ for $i=3,4$, ${\mathrm{RM}}_4(3m-i,m)$ for $i=2,3$, ${\mathrm{RM}}_5(4m-4,m)$, and ${\mathrm{RM}}_7(6m-4,m)$, in the ranges of $m$ specified in the corresponding results. The proofs proceed by induction on $m$, using a sumset inclusion for GRM weight spectra together with results on low-weight codewords. The required base cases are established by computations in Magma and, for certain ternary cases, exact constraint solving with Z3.