We construct infinite families of binary or ternary linear complementary dual (LCD) codes, which arise from some families of self-orthogonal codes associated with simplicial complexes or multi-variable functions. We find explicit criteria for self-orthogonal codes to preserve self-orthogonality after augmentation. For the construction of LCD codes, we either directly use some families of self-orthogonal codes or we use the augmented codes of these self-orthogonal code families. As a result, we obtain several infinite families of binary or ternary LCD codes, which include LCD-optimal families and at least LCD-almost optimal families; we also present their weight distributions. We mention that our LCD code families are new compared with the data summarized in [27, Table 5] based on the previously known LCD code families.
We describe a new, highly optimized implementation of number theoretic transforms on processors with SIMD support (AVX, AVX-512, and Neon). For any prime modulus p and any order of the form r = 2^i · 3^j |p - 1 , our implementation can automatically generate a dedicated codelet to compute the number theoretic transform of order r over 𝔽_p . New speed-ups were achieved by relying heavily on non-normalized modular arithmetic and allowing for orders r that are not necessarily powers of two.
We investigate m-adic residue codes of prime length p over the mixed-alphabet ring 𝔽_q S, where S=𝔽_q+v𝔽_q with v^2=v, assuming q is an odd prime power with (p,q)=1 and m| (p-1). Using the structural decomposition 𝔽_q S ≅𝔽_q^3, we describe cyclic codes as principal ideals generated by lifted orthogonal idempotents and classify them into even-like and odd-like families of Class I and Class II, with explicit generators, orthogonality relations, and size formulas. A Gray map from 𝔽_q S to 𝔽_q^3 is introduced, together with a Lee weight and a Gray-induced bilinear form, yielding Lee distance bounds in terms of the three cyclic constituents over 𝔽_q. At the idempotent level, we determine dual codes by analyzing the action of the reciprocal automorphism (and, when q=r^2, the Frobenius automorphism), and derive a combinatorial disjoint-support criterion on index sets that guarantees the required dual containment for both Euclidean and Hermitian inner products. This leads to a unified framework for constructing Euclidean and Hermitian CSS-type quantum stabilizer codes over 𝔽_q, with parameters expressed in terms of the degrees of the underlying generator polynomials. Explicit examples illustrate the resulting classical and quantum code families.
Let R be a ring with involution * . The * -symmetric graph of R is a simple graph with vertex set as the set of all nonzero zero-divisors of R and two distinct vertices x and y are adjacent if xy=0 or yx=0 and yx^*=0 . The * -symmetric graph is a generalization of the well known zero-divisor graph of R. In this paper, we investigate the interplay between the ring-theoretic properties of * -ring R and the graph-theoretic properties of * -symmetric graph of R.
Minimal linear codes are of interest due to their wide application in secret sharing schemes and secure two-party computation. In this paper, we construct several families of binary minimal linear codes from generic construction. In this paper, their weight distributions are determined, and the sufficient conditions for them to be minimal are given. It is worth noting that some minimal binary linear codes obtained in this paper violate the Ashikhmin–Barg condition and can be used to design a secret sharing scheme with a good access structure.
For unital and commutative algebras over an algebraically closed field, 𝕂 , any inclusion of finite codimension can be characterised as a chain of subalgebra inclusions of codimension 1. Each such inclusion A ⊂ B can also be interpreted as a linear condition that holds on A but not on the whole of B. The set of conditions above A in the chain are called the subalgebra conditions of A. We investigate the behaviour of such chains (or sets of conditions) when the said algebras are ideal subalgebras. That is, when they are sums of the base field and an ideal. We then move to the setting of subalgebras of the univariate polynomial ring, where we can give a more concrete description of subalgebra conditions. We consider a wider class of so-called single-clustered polynomial subalgebras. We show that any subalgebra in 𝕂[x] of finite codimension is a finite intersection of single-clustered subalgebras. We then present an algorithm to compute subalgebra conditions of single-clustered subalgebras from generators using linear methods. This method is a generalisation of a method used for almost monomial subalgebras, that is, subalgebras with only one element in the spectrum.
We describe the notion of independence and modular independence of vectors and count left and right linear codes over the non-commutative, non-unital, ring E of order 4. Moreover, we produce a Gray map which gives a bijection between right linear codes of length n and binary linear codes of length 2n.
This work investigates the problem of identifying admissible paths in a tree T to provide a complete characterization of a reduced Gröbner basis 𝒢 of the binomial edge ideal of T. To this end, we develop efficient algorithms that compute all admissible paths in G without exhaustively generating and checking every possible path. We explore three labeling strategies based on classical graph traversal methods, Depth-First Search (DFS), Breadth-First Search (BFS), and inorder traversal for rooted binary trees. For each labeling, we find admissible paths, present optimized algorithms to compute them, and analyze their computational complexity. In the DFS context, we demonstrate that any tree T is m-closed for some m ≤rad(T) + 2 , where rad(T) is the radius of the tree. Via inorder traversal, we show that every rooted binary tree of height h is m-closed for some m ≤ h + 1 . These results contribute to the structural understanding of binomial edge ideals and their Gröbner bases under various graph labelings.
In this paper we continue the work of describing polynomial subalgebras of finite codimension that was started in Grönkvist et al. (Appl Algebra Eng Commun Comput 33(6):751–789, 2022). Let 𝕂 be an algebraically closed field, and A ⊂𝕂[x_1, … , x_n] be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of 𝕂 -algebras A = A_0⊂ A_1⊂⋯⊂ A_m = 𝕂[x_1, … , x_n], where each A_i can be written as the kernel of some linear functional L_i + 1: A_i + 1→𝕂 , and each L_i is either a derivation or of the form L_i: f → c(f(α) - f(β)) for some α, β∈𝕂^n and c ∈𝕂 . We investigate the structure of these filtrations and linear functionals. Our main result shows that each such L_i which is a derivation may be written as a linear combination of partial derivatives evaluated at points of 𝕂^n .
This paper aims to define linear Gray functions over the Frobenius non-chain ring GF(2(d))[X,Y,Z]/< X-2, Y-2, YZ, XZ, Z(2)-XY >. Some Gray functions map linear codes to codes consisting of 2n transpositions, while others map linear codes to codes with an automorphism consisting of n transpositions. Self-orthogonal constacyclic codes over rings whose maximal ideals have nilpotency index three are thoroughly described, where the length of the code is relatively prime to the characteristic of the residue field of the ring. Hence, these codes over our ring are determined, and their Gray images are discussed.
Low-Rank Parity check codes over finite fields have gained a lot of attention since their introduction in 2013 by Gaborit et al. as a new family of rank metric codes due to their application in cryptography particularly. After the definition of rank metric codes over finite principal ideal rings by Kamche et al., several works have generalized Low-Rank Parity check codes over finite commutative rings using notions in module theory. This rank metric is related to the number of elements in a minimal generating family of a module. Epelde et al. have introduced a new metric over Galois rings, taking into account the cardinality of the module and defined Gabidulin codes in this context. In this paper, we give the definition of Low-Rank Parity check codes with this new metric. We study some properties of the product of two submodule with the cardinal rank metric and derive the success probability of the decoder. We then compare this metric with the rank metric showing the advantages of the rank metric.
Following results and ideas due to J. Pawlina and Ş. O. Tohăneanu we consider lower bounds for the minimum distances of an evaluation code obtained evaluating all degree a forms (or the forms in a subspace) at the points in a finite subset X of a projective space. We handle some cases when there are degree a forms vanishing on X. We also consider the codes obtained evaluating X at the subspace defined by a zero-dimensional scheme Z such that Z∩ X=∅ . These codes arise from multiple-point codes of embedded curves.
In this paper, we study the weighted Fermat-Frechet problem for a N (N+1)/2 -tuple of positive real numbers determining N-simplexes or an N-simplex in the N-dimensional K-space (N-dimensional Euclidean space ℝ^N if K=0 , the N-dimensional open hemisphere of radius 1/√(K) ( 𝕊_1/√(K)^N ) if K >0 and the Lobachevsky space ℍ_K^N of constant curvature K if K<0 ). The (weighted) Fermat-Frechet problem is a new generalization of the (weighted) Fermat problem for N-simplexes. We control the number of solutions (weighted Fermat trees) with respect to the weighted Fermat-Frechet problem that we call a weighted Fermat-Frechet multitree, by using some conditions for the edge lengths discovered by Dekster–Wilker. We use the isometric immersion of Godel-Schoenberg for N-simplexes in the N-sphere and the isometric immersion of Gromov (up to an additive constant) for weighted Fermat (Steiner) trees in the N-hyperbolic space ℍ_K^N, in order to construct an isometric immersion of a weighted Fermat-Frechet multitree in the K-space. Finally, we create a new variational method, which differs from Schlafli’s, Luo’s and Milnor’s techniques to differentiate the length of a geodesic arc with respect to a variable geodesic arc, in the 3K-space. By applying this method, we eliminate one variable geodesic arc from a system of equations, which gives the weighted Fermat-Frechet solution for a sextuple of edge lengths determining (Frechet) tetrahedra.
Boolean functions and binary sequences are fundamental tools in cryptography. In this work, we introduce a new bijection between the set of Boolean functions and the set of binary sequences whose period is a power of two. This correspondence enables the study of properties of Boolean functions through binary sequences and vice versa. Building on this connection, we propose a novel algebraic description, derived from the algebraic normal form of Boolean functions, which we call the reverse-ANF. Then, we explore how this formulation relates both to existing representations of Boolean functions and to binary sequences. Moreover, several cryptographic properties are examined through this new approach. Finally, we analyse generalized self-shrunken sequences from the perspective of Boolean functions, highlighting several properties that emerge under these different frameworks.
This paper aims to define linear Gray functions over the Frobenius non-chain ring GF(2^d)[X,Y,Z]/⟨ X^2, Y^2, YZ, XZ, Z^2-XY⟩ . Some Gray functions map linear codes to codes consisting of 2n transpositions, while others map linear codes to codes with an automorphism consisting of n transpositions. Self-orthogonal constacyclic codes over rings whose maximal ideals have nilpotency index three are thoroughly described, where the length of the code is relatively prime to the characteristic of the residue field of the ring. Hence, these codes over our ring are determined, and their Gray images are discussed.
We study a non-unital and non-commutative ring S-m(R), called ring of ordered sum over a ring R. We obtain linear codes over S-m(F-2), also known as S-m-codes, where F-2 is the binary field. The algebraic structure of S-m-codes, particularly, their residue and torsion codes, will be explored. Moreover, a generalized notion of quasi self-dual codes will be introduced. Finally, we give results on the weight enumerators of these codes and construct S-3-codes in short lengths using the residue and torsion codes.
