A gap in GRM code weight distributions (Q1970569)

The weight distribution of generalized Reed-Muller (GRM) codes is open, except for some particular parameters. This paper contributes to the research on this problem by proving results on weights that cannot occur in GRM codes. Every \([n,k,d]\) GRM code has words of weight 0, \(d\), and \(n\), and obviously no words of weight between 0 and \(d\). Let \(\Delta\) be the largest integer dividing all weights occurring in a code. Then there are no words having weight between \(d + i\Delta\) and \(d+(i+1)\Delta\), \(i \geq 0\). This paper shows that, for some GRM codes, the gap is actually larger: there are no words having weight between \(d\) and \(d+2\Delta\).