We investigate the random generation of Latin rectangles using a method based on the assignment problem. Specifically, each row is selected via a minimum-cost permutation from a randomly generated cost matrix. This approach defines a convex polytope for each Latin rectangle, where the volume of the polytope determines the sampling probability of the corresponding rectangle. Our analysis reveals that the resulting process is efficient but generally non-uniform, thereby providing a negative answer to a question posed by the second author in 2009. Furthermore, we establish that the volume of the polytope is invariant under column and symbol permutations of the rectangles.

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.

Abstract It is known that correlation-immune (CI) Boolean functions used in the framework of side channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays, when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of rows in a simple orthogonal array. In this paper, we use Rao’s Bound to give a sufficient condition on the number of rows, for a binary orthogonal array (OA) to be simple. We apply this result for determining the minimum number of rows in all simple binary orthogonal arrays of strengths 2 and 3; we show that this minimum is the same in such case as for all OA, and we extend this observation to some OA of strengths 4 and 5. This allows us to reply positively, in the case of strengths 2 and 3, to a question raised by the first author and X. Chen on the monotonicity of the minimum Hamming weight of 2-CI Boolean functions, and to partially reply positively to the same question in the case of strengths 4 and 5.

Techniques of producing new combinatorial structures from old ones are commonly called trades. The switching principle applies for a broad class of designs: it is a local transformation that modifies two columns of the incidence matrix. In this paper, we present a construction, which is a generalization of the switching transform for the class of Steiner 2-designs. We call this construction paramodification of Steiner 2-designs, since it modifies the parallelism of a subsystem. We study in more detail the paramodifications of affine planes, Steiner triple systems, and abstract unitals. Computational results show that paramodification can construct many new unitals.

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).