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 Sørensen 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.

On the next-to-minimal weight of projective Reed–Muller codes

“…We summarize the results for W (2) PRM (n, d) obtained in [3] and in this paper in the following tables, where we also list the corresponding values of W…”

“…In [3] we determined all values for the next-to-minimal weights of projective Reed-Muller codes defined over F 2 , and from those results we get that there are cases for which equality does not hold in (2.2). In the next section we will determine several values for the next-to-minimal weights of projective Reed-Muller codes defined over F q , with q ≥ 3, and we will also find some cases where equality does not hold in (2.2).…”

“…In the next section we will determine several values for the next-to-minimal weights of projective Reed-Muller codes defined over F q , with q ≥ 3, and we will also find some cases where equality does not hold in (2.2). We recall from [3] a definition and some results that we will need to prove the main results.…”

“…As for the determination of the next-to-minimal weight for these codes, there are some results (also about higher Hamming weights) on this subject by Rodier and Sboui ([14], [15], [18]) and also by Ballet and Rolland ([1]). Recently the authors of this note completely determined the next-to-minimal weight of projective Reed-Muller codes defined over F 2 (see [3]) In this paper we present several results on the next-to-minimal Hamming weights for projective Reed-Muller codes, including the complete determination of the next-to-minimal weights for the case of projective Reed-Muller codes defined over F 3 . In the next section we recall the definitions of the generalized and projective Reed-Muller codes, and some results of geometrical nature that will allow us to determine many cases of higher Hamming weights for the projective Reed-Muller codes, which we do in the last section.…”

Determining the Generalized Hamming Weight Hierarchy of the Binary Projective Reed-Muller Code

“…To the best of our knowledge, none of the previous works on GHW of PRM codes have considered the binary (q = 2) version. However, next-to-minimal weight of binary PRM codes is determined in a recent work [12], wherein next-to-minimal weight means minimal codeword weight that is greater than minimum Hamming weight. Note that the next-to-minimal weight is not the same as second generalized Hamming weight.…”