Weighted Reed–Muller codes revisited

Abstract

We consider weighted Reed-Muller codes over point ensemble S1 × · · · × Sm where Si needs not be of the same size as Sj. For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S1|/|S2| on the minimum distance. In conclusion the weighted Reed-Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed-Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.

Minimum distance functions of complete intersections

“…The basic parameters of the projective Reed-Muller-type code C X (d) are equal to those of C X * (d) [22]. A formula for the minimum distance of an affine cartesian code is given in [21, Theorem 3.8] and in [14,Proposition 5]. A short and elegant proof of this formula was given by Carvalho in [5, Proposition 2.3], where he shows that the best way to study the minimum distance of an affine cartesian code is by using the footprint.…”

Generalized Hamming weights of affine Cartesian codes

“…Also in [10] it was observed that the answer to Question 1.1 for the case r = 1 gives the minimum distance of what they called affine cartesian codes. It was brought to our attention by Olav Geil that, these codes were already studied in [7] in a more general setting and the answer to Question 1.1 for r = 1 is a special case of [7,Prop. 5].…”

Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes

“…An affine Cartesian code of order r is a monomial-Cartesian code C(S, A r ), where A r is as above and S is an arbitrary Cartesian set. This family of codes appeared first time in [20] and then independently in [28]. In [20], the authors study the basic parameters of Cartesian codes, they determine optimal weights for the case when A r is the Cartesian product of two sets, and then present two list decoding algorithms.…”