Abstract The Datta–Johnsen code is an evaluation code where the linear combinations of elementary symmetric polynomials are evaluated on the set of all points with pairwise distinct coordinates in an affine space of dimension $$\ge \, 2$$ ≥ 2 over a finite field $$\mathbb {F}_q$$ F q . A generalization is obtained by taking a low dimensional linear system of symmetric polynomials. The odd characteristic case was the subject of a recent paper. Here, the even characteristic case is investigated.

Abstract A problem of current interest, also motivated by applications to Coding theory, is to find explicit equations for maximal curves, that are projective, geometrically irreducible, non-singular curves defined over a finite field $$\mathbb {F}_{q^2}$$ F q 2 whose number of $$\mathbb {F}_{q^2}$$ F q 2 -rational points attains the Hasse-Weil upper bound $$q^2+2\mathfrak {g}q+1$$ q 2 + 2 g q + 1 where $$\mathfrak {g}$$ g is the genus of the curve $$\mathcal {X}$$ X . For curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order and also when has order the square of the characteristic. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order dp where p is the characteristic of $$\mathbb {F}_{q^2}$$ F q 2 and d is a prime other than p. We also compute the generators of the Weierstrass semigroup at a special $$\mathbb {F}_{q^2}$$ F q 2 -rational point of some of the curves, and discuss some possible positive impacts on the minimum distance problems of AG-codes.

Abstract Datta and Johnsen (Des Codes Cryptogr 91:747–761, 2023) introduced a new family of evaluation codes in an affine space of dimension $$\ge 2$$ ≥ 2 over a finite field $${\mathbb {F}}_q$$ F q 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 $$q=7,9$$ q = 7 , 9 shows that carefully chosen generalized Datta–Johnsen codes $$\left[ \frac{1}{2}q(q-1),3,d\right] $$ 1 2 q ( q - 1 ) , 3 , d have minimum distance d equal to the optimal value minus 1.