Let m, u be positive integers and let Fq denote the finite field with q elements. Affine Cartesian codes are linear error-correcting codes obtained from evaluating polynomials in m variables of degree ≤ u over a Cartesian product of the form A1 × · · · × Am, where A1, . . . , Am are subsets of Fq . These codes were first discussed by Geil and Thomsen in 2013 in a more general setting and were formally introduced by Lopez, Renteria-Marquez and Villarreal in 2014. Reed-Muller codes and Reed-Solomon codes can be considered special cases of affine Cartesian codes. In 1970, Delsarte, Goethals and MacWilliams gave a characterization of minimum weight codewords of Reed-Muller codes and also enumerated them. Extending the work of Delsarte et al., Carvalho and Neumann in 2020 considered affine Cartesian codes in a special setting where they considered the sets A1, . . . , Am to be nested subfields of Fq and gave a characterization of its minimum weight codewords. We give an explicit formula for the number of minimum weight codewords in this special case of affine Cartesian codes. As a corollary, we deduce the known formulas for the number of minimum weight codewords of Reed-Solomon codes and of Reed-Muller codes. 

This is a joint work with Sudhir R. Ghorpade.