Title: The next-to-minimal weights of binary projective Reed-Muller codes,
Abstract: Projective Reed-Muller codes were introduced by Lachaud, in 1988 and their
dimension and minimum distance were determined by Serre and S{\o}rensen in
1991. In coding theory one is also interested in the higher Hamming weights, to
study the code performance. Yet, not many values of the higher Hamming weights
are known for these codes, not even the second lowest weight (also known as
next-to-minimal weight) is completely determined. In this paper we determine
all the values of the next-to-minimal weight for the binary projective
Reed-Muller codes, which we show to be equal to the next-to-minimal weight of
Reed-Muller codes in most, but not all, cases.