Evaluation codes arising from symmetric polynomials

Datta and Johnsen (Des Codes Cryptogr 91:747–761, 2023) introduced a new family of evaluation codes in an affine space of dimension ≥ 2 over a finite field Fq where linear combinations of elementary symmetric polynomials are evaluated on the set of all points with pairwise distinct coordinates. In this paper, we propose a generalization by taking low dimensional linear systems of symmetric polynomials. Computation for small values of q = 7, 9 shows that carefully chosen generalized Datta–Johnsen codes [1/2q(q-1), 3, d have minimum distance d equal to the optimal value minus 1.