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 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.

Hermitian functional and differential codes are AG-codes defined on a Hermitian curve. To ensure good performance, the divisors defining such AG-codes have to be carefully chosen, exploiting the rich combinatorial and algebraic properties of the Hermitian curves. In this paper, the case of differential codes CΩ(D, mT) on the Hermitian curve ℋq3 defined over Fq6 is worked out where su.yppp(T) := ℋq3(Fq2), the set of all Fq2-rational points of ℋq3, while D is taken, as usual, to be the sum of the points in the complementary set D = ℋq3(Fq6) \ℋq3(Fq2). For certain values of m, such codes CΩ(D, mT) have better minimum distance compared with true values of 1-point Hermitian codes. The automorphism group of CL(D, mT), m ≤ q3- 2, is isomorphic to P GU(3, q).