On the next-to-minimal weight of affine cartesian codes

In this paper we determine many values of the second least weight of codewords, also known as the next-to-minimal Hamming weight, for a type of affine variety codes, obtained by evaluating polynomials of degree up to d on the points of a cartesian product of n subsets of a finite field Fq. Such codes firstly appeared in a work by O. Geil and C. Thomsen (see [12]) as a special case of the so-called weighted Reed-Muller codes, and later appeared independently in a work by H. López, C. Rentería-Marquez and R. Villarreal (see [16]) named as affine cartesian codes. Our work extends, to affine cartesian codes, the results obtained by Rolland in [17] for generalized Reed-Muller codes.