Holes in Generalized Reed–Muller Codes

The possible relative weights of codewords of Generalized Reed-Muller codes are studied. Let
RMq
(
r
,
m
) denote the code of polynomials over the finite field
Fq
in
m
variables of total degree at most
r
. The relative weight of a codeword
f
¿ RM
q
(
r
,
m
) is the fraction of nonzero entries in
f
. The possible relative weights are studied, when the field
Fq
and the degree
r
are fixed, and the number of variables
m
tends to infinity. It is proved that the set of possible weights is sparse-for any
¿
which is not rational of the form
¿ = ¿/q

^k
, there exists some
¿ > 0
such that no weights fall in the interval
(¿-¿,¿+¿)
. This demonstrates a new property of the weight distribution of Generalized Reed-Muller codes.