Journal of Algebra Combinatorics Discrete Structures and Applications

Multidecomposition of complete graphs into cycles and claws

2026-05-06T12:12:00+03:00

Let C_n and S_n respectively denote a cycle and star with n edges. Let K_n denote a complete graph on n vertices. In this paper, it is shown that for any non-negative integers α and β and any positive integer n ≥ 6, there exists a decomposition of K_n into α copies of C₆ and β copies of S₃ if and only if

6α + 3β = n(n − 1) 2

β ≠ 1, 2 when n is odd, and β ≥ ⌈n/4⌉ when n is even.

A note on the algebra of threshold graphs

2026-05-06T12:11:59+03:00

It is known that threshold graphs have edge rings with 2-linear resolutions. This was proved by Engström and Stamps, [4]. They used the fact that an edge ring of a graph G has a 2-linear resolution if and only if the complement graph is chordal. They also described a method to determine the Betti numbers. Our goal is to determine when edge rings of threshold graphs are Cohen-Macaulay. In order to do so, it is more convenient to use an alternative way to study edge rings of graphs, that is to interpret them as Stanley-Reisner rings. We also determine when the neighborhood complex of a threshold graph has a Cohen-Macaulay Stanley-Reisner ring.

On the spectrum of some cyclically and transitively oriented H-designs

2026-05-06T12:11:58+03:00

This paper investigates oriented ℱ-designs on complete uniform hypergraphs of rank 3, focusing in particular on the spectrum of existence and on the construction of some cyclically and transitively oriented P⁽³⁾(2, 4)-designs and P⁽³⁾(1, 5)-designs, namely BCP⁽³⁾(2, 4)-designs and BCP⁽³⁾(1, 5)-designs for cyclically oriented ones, BTP⁽³⁾(2, 4)-designs and BTP⁽³⁾(1, 5)-designs for transitively oriented ones. In the appendix, we provide the Python code to obtain the explicit realization of the BCP⁽³⁾(2, 4)-designs on v vertices. Moreover, the structure of this algorithm, with suitable modifications, can be generalized to the other three structures as well.

Some constructions of unimodular lattices via totally real subfields of the p-th cyclotomic field

2026-05-06T12:11:57+03:00

Algebraic number theory has recently attracted significant interest due to its role in algebraic lattice theory and in the design of codes for applications in coding theory. Algebraic lattices have been useful in information theory, where the problem of constructing lattices over number fields with full diversity and maximal minimum product distance has been investigated, since these parameters are directly related to error probabilities over Rayleigh fading channels. In this paper, we present a family of full diversity rotated unimodular lattices constructed via totally real subfields of the cyclotomic fields ℚ(ζ_p), with p an odd prime. A closed-form expression for the minimum product distance is derived.

Quaternions over Galois rings and their codes

2026-05-06T12:11:57+03:00

It is shown that, if R is a Frobenius ring, then the quaternion ring ℱ_a,b(R) is a Frobenius ring for all units a, b ∈ R. In particular, if q is an odd prime power, then ℱ_a,b(F_q) is the semisimple non-commutative matrix ring M₂(F_q). Consequently, a homogeneous weight that depends on the field size q is obtained. Moreover, the homogeneous weight of a finite Frobenius ring with a unique minimal ideal is derived in terms of the size of the ideal. This is illustrated by the quaternions over the Galois ring GR(2^r, m). Finally, one-sided linear block codes over the quaternions with coefficients in the Galois ring are constructed, and certain bounds on the homogeneous distance of the images of these codes are proved. These bounds are based on the Hamming distance of the quaternion code and the parameters of the Galois ring. Good examples of one-sided rate-2/6, 3-quasi-cyclic quaternion codes and their images are generated. One of these codes meets the Singleton bound and is therefore a maximum distance separable code.

Generalized subspace subcodes in the rank metric

2026-05-06T12:11:55+03:00

Rank-metric codes were studied by E. Gabidulin in 1985 after a brief introduction by Delsarte in 1978 as analogues of Reed-Solomon codes in the rank metric, but based on linearized polynomials. They have found applications in many areas, including linear network coding and space-time coding.

They are also used in cryptography to reduce the size of the keys compared to Hamming metric codes at the same level of security. However, some families of rank-metric codes suffer from structural attacks due to the strong algebraic structure from which they are defined.

It therefore becomes interesting to find new code families in order to address these questions in the landscape of rank-metric codes. In this paper, we provide a generalization of Subspace Subcodes in rank metric introduced by Gabidulin and Loidreau. We also characterize this family by giving an algorithm which allows one to obtain its generator and parity-check matrices based on the associated extended codes. We have also studied the specific case of Gabidulin codes whose underlying decoding algorithms are known. Bounds for the cardinalities of these codes, both in the general case and in the case of Gabidulin codes, are also provided.

Expansion of implicit functions into formal power series in terms of partial Bell polynomials

2026-05-06T12:11:54+03:00

Starting from the representation of a function f(x, y) as a formal power series with Taylor coefficients f_m,n, a formal series is set up for the implicit function y = y(x) so that f(x, y) = 0 and the coefficients of the series for y depend exclusively on the f_m,n. The solution to this problem provided here relies on using partial Bell polynomials and their inverse companions. Some examples and applications are discussed.

EAQEC codes from polycyclic codes over the ring R_l

2026-05-06T12:11:52+03:00

In this paper, we study polycyclic codes over the ring R_l = F_q[w] / 〈w^l − 1〉 with q = p^k, where k is a positive integer and p is an odd prime. We explore LCD annihilator, self-dual, self-orthogonal codes over R_l and polycyclic codes over R_l. Moreover, we provide a structure of entanglement-assisted quantum error-correcting codes based on the developed polycyclic codes through the inclusion of dual contained conditions. Subsequently, some LCD hull code-derived entanglement-assisted quantum error-correcting code examples are presented.

Sieving for large twin smooth integers using single solutions to Prouhet-Tarry-Escott

2026-05-06T12:11:52+03:00

In the isogeny-based track of post-quantum cryptography, optimal instances of the signature scheme SQISign rely on primes p such that p ± 1 is smooth. In 2021, a new approach to find those numbers was discovered using solutions to the Prouhet-Tarry-Escott (PTE) problem. With these solutions, we can sieve for smooth integers A and B with a difference of |A − B| = C fixed by the solution. Then some 2A/C and 2B/C are smooth integers hopefully enclosing a prime.

They took many different PTE solutions and combined them into a tree to process them more efficiently. But for larger numbers, there are fewer promising PTE solutions, so their advantage over the naive approach, namely checking a single solution at a time, fades. For a single PTE solution, the search can be optimized for the corresponding C and allows one to check smoothness only for those integers that are divisible by C. In this work, we investigate such optimisations and show a significant speed-up compared to the naive approach, both heuristically and empirically. Along the way, we compute the number of roots of a given polynomial modulo prime powers and give an upper bound for the number of roots modulo a composite number.

Skew cyclic codes over a finite non-chain ring and an application

2026-05-06T12:11:49+03:00

This article studies Θ_t-cyclic and (Θ_t, λ)-constacyclic codes over the finite commutative non-chain Frobenius ring R = F_q[u, v, w] / 〈 u² − u, v² − v, w² − 1, uv, uw − wu, wv − vw 〉. Gray maps, structural decompositions, and generator descriptions are developed for both odd- and even-characteristic cases. The paper further determines principal generators in the associated skew polynomial rings, dual codes, idempotent generators, and conditions for self-duality. It also presents explicit examples over specific finite fields and extends the framework to DNA codes in the even-characteristic setting through reversibility and complement constraints. Spanning sets, cardinality formulas, and optimal DNA-code constructions meeting the Griesmer bound are also obtained.