Butson Hadamard matrices are complex Hadamard matrices with entries in the complex roots of unity of given order. There is an interesting code in phase space related to these matrices called Butson Hadamard codes in (Armario et al. 2023). We study the covering radius of these codes for the homogeneous weight, a weight of fundamental importance in codes over rings. It is defined uniquely, up to scaling, for a commutative ring alphabet that is Quasi Frobenius. Two upper bounds on the covering radius are derived by an orthogonal array argument. A lower bound relies on the existence of bent sequences in the sense of (Shi et al. 2022). This latter bound generalizes a bound of (Armario et al. 2025) for the Hamming weight.

Let Φ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann Ξ-function, set s(t)=Φ(√(t)), and define the first Laguerre expression f(t)=s'(t)^2-s(t)s”(t). Csordas and Dimitrov (2000) conjectured that log f is strictly concave on (0,∞); Csordas (2015) later restated the assertion as Open Problem 4.14. We prove the conjecture by two complementary methods, sharing only a local certificate near the modular fixed point. The first proof is a direct theta-series argument: a directed-rounding Taylor certificate near t=0 is joined to a dominant-first-summand estimate with rigorous theta-tail bounds. The second proof uses Jacobi's nonlinear third-order differential equation for θ_3 to obtain a three-dimensional autonomous phase space, a sharp elliptic monotonicity theorem, and an exact quartic reduction of the target inequality. The quartic boundary is a polynomial shear of the quadratic cone XY=Z^2. A directed interval certificate proves that every possible cone contact on the only remaining compact interval points strictly into the desired region; beyond that interval a pointwise monotonicity theorem closes the argument. The finite certificates and exact symbolic checks are supplied as reproducible scripts. The result implies the associated double Turán inequalities through the theorem of Csordas–Dimitrov, but no assertion of the Riemann Hypothesis is made.

The punctured dodecacode is an additive 4-ary code of length 11 and distance 5 which is uniformly packed. We show that a code with the same weight distribution is equivalent to it. This code is also shown to be nonlinear. We also establish the nonexistence of analogues of the dodecacode and the punctured dodecacode in Doob graphs. To that end, we classify two-weight codes of weights 6 and 8 in Doob and 4-ary Hamming graphs of diameter 9 and the corresponding strongly regular graphs. Keywords: dodecacode, additive code, trace Hermitian duality, uniformly packed code, completely regular code, Doob graph, strongly regular graph

In this paper, we propose a novel multisecret sharing (MSS) scheme that integrates a recently developed exponential-speedup matrix spectral factorization algorithm into the construction of paraunitary matrices over finite fields. By exploiting the block-matrix generalization of the Janashia-Lagvilava method, we significantly enhance the efficiency and scalability of the MSS scheme. The proposed method ensures perfect secrecy, collusion resistance, and efficient reconstruction, while enabling practical deployment in large-scale distributed systems such as secure cloud storage, IoT networks, and blockchain authentication. Security and performance analyses demonstrate the superiority of the new approach over existing MSS schemes.

Wall-Sun-Sun primes (shortly WSS primes) are defined as those primes p such that the period of the Fibonacci recurrence is the same modulo p and modulo p^2. This concept has been generalized recently to certain second order recurrences whose characteristic polynomials admit as a zero the principal unit of ℚ(√(d)), for some integer d>0. Primes of the latter type we call WSS(d). They correspond to the case when ℚ(√(d)) is not p-rational. For such a prime p we study the weight distributions of the cyclic codes over 𝔽_p and ℤ_p^2 whose check polynomial is the reciprocal of the said characteristic polynomial. Some of these codes are MDS (reducible case) or NMDS (irreducible case).

We generalize the fundamental bounds of Delsarte thesis (1973) on codes of given degree and designs of given strength in the new setting of Bannai et al. (2025). We assume the scheme is weakly metric in the sense of (Solé, 1989). We give upper bounds on the size of codes of given degree, and also on the size of codes with a given number of pairwise distances. Codes meeting these bounds are characterized by the identification of suitable annihilators with the degree (resp. distance) Wilson polynomial. We give two analogues of the Rao bound on the size of designs with given strength. Designs meeting that bound we call degree tight designs or distance tight design depending on the bound met. In both cases, the existence of a tight design implies a Lloyd-like condition on a suitable analogue of the Wilson polynomial. Applications to the Lee distance, mixed level orthogonal arrays, ordered orthogonal arrays, and more are given. The formal duality between codes and designs, connecting perfect codes and tight designs, is made concrete in self-dual translation schemes.

Group codes form an important class of spherical codes, consisting of a single orbit under a subgroup of the orthogonal group. They have been studied by (Mittelholzer-Lahtonen, 1996) in the case of Coxeter groups for their packing radius. We study them here for the same groups, with respect to their covering radius. An exact algorithm to determine their covering radii is derived, based on geometric ideas, and applied to tabulate their values in dimensions up to 8. The values of the covering radii are compared to the sphere covering bound, and to the bounds in (Fazekas-Levenshtein, 1995) and (Boyvalenkov-Stoyanova, 2021) on the covering radius of spherical designs with given strength. For many sizes of codes, our codes are reasonably sparse sphere coverings, and the only known in these dimensions.

We introduce the notion of a bent sequence attached to a weighing matrix. With every weighing matrix, we associate a spherical code in dimension twice the order of the matrix. Its covering and packing properties are studied, and under mild conditions, it is shown to be a spherical design of strength two. The bent sequences provide a lower bound, and the design property an upper bound on the covering radius of the code. Exact values are determined by using a geometric technique (Delaunay diagram). Computational methods based on Gr & ouml;bner bases and linear algebra are given to construct bent sequences. Balancedly splittable matrices provide constructions of weighing matrices with designed bent sequences. (c) 2025 Published by Elsevier B.V.

In this paper, we explore the hull of linear codes over the non-unitary rings of order four, namely E, I and H. Initially, we determine the binary associated codes of the hull over each ring; then, we characterize the hulls in terms of these codes. Furthermore, we establish the connection between the hull of any linear code over these rings and the hull of its dual. Finally, we classify linear codes over each ring with prescribed hull sizes under permutation equivalence for short lengths.

A near-field is an algebraic structure akin to a division ring where distributivity is relaxed on one side. The smallest such object is the Dickson near-field of order nine. We lay the foundations of linear codes over that near-field. This includes a systematic form for their generator matrices based on a one-sided Gauss pivot algorithm. Dual codes are defined and a formula is given for the parity check matrix. LCD codes are characterized by a direct sum property. Self-orthogonal codes are classified in short lengths.

In this study, we consider the Euclidean and Galois hulls of multi-twisted (MT) codes over a finite field FPe of characteristic p. Let G be a generator polynomial matrix (GPM) of an MT code C. For any 0 <= K < e, the t-Galois hull of C, denoted by h(C), is the intersection of C with its t-Galois dual. The main result in this paper is that a GPM for h(C) has been obtained from G. We start by associating a linear code Q(G) with G. We show that Q(G) is quasi-cyclic. In addition, we prove that the dimension of h(K)(C) is the difference between the dimension of C and that of Q(G). Thus the determinantal divisors are used to derive a formula for the dimension of h(K)(C). Finally, we deduce a GPM formula for h(K)(C). In particular, we handle the cases of t-Galois self-orthogonal and linear complementary dual MT codes; we establish equivalent conditions that characterize these cases. Equivalent results can be deduced immediately for the classes of cyclic, constacyclic, quasi-cyclic, generalized quasi-cyclic, and quasi-twisted codes, because they are all special cases of MT codes. Some numerical examples, containing codes with the best-known parameters, are used to illustrate the theoretical results. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A lattice is B-modular for B an integer if it is equivalent to its dual rescaled by root B. A notion of strongly modular lattice exists for B = 6, 14, 15. When B = 15 a construction of modular lattices is given from Type IV codes over F2 & times; F2 regarded as a quotient of the ring of integers of a quadratic number field by a principal ideal. The theta series is obtained by substituting some special weight 1 modular forms into the symmetrized weight enumerator of the code. One of these forms occurs in Ramanujan's theory of elliptic functions to the base 15 as developed by Cooper and Ye (2015). We give a map between the ring of polynomial invariants for a certain matrix group of dimension 3 and a subring of the ring of modular forms where the theta series of 15-modular lattices live, thus solving, in a weak sense, an open problem of Choie and Sol & eacute; (2008). Using the coding approach, we give an alternate proof of a theta series identity due to Chan, Chua, and Sol & eacute; (2003). We classify optimal self-dual Type IV codes over F2 & times; F2 in short lengths, thus addressing the open problems proposed by Dougherty, Gaborit, Harada, Munemasa, and Sol & eacute; (1999), and construct the attending 15-modular lattices. (c) 2026 Published by Elsevier Inc.

The size of the Hamming distance spectrum of a code has received great attention in recent research. The main objective of this paper is to extend these significant theories to the b-symbol distance spectrum. We examine this question for various types of codes, including unrestricted codes, additive codes, linear codes, and cyclic codes, successively. For the first three cases, we determine the maximum size of the b-symbol distance spectra of these codes smoothly. For the case of cyclic codes, we introduce three approaches to characterize the upper bound for the cardinality of the b-symbol weight spectrum of cyclic codes, namely the period distribution approach, the primitive idempotent approach, and the b-symbol weight formula approach. As two by-products of this paper, the maximum number of symplectic weights of linear codes is determined, and a basic inequality among the parameters [n, k, dH(C)]q of cyclic codes is provided. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The Lloyd Theorem of (Solé, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of polynomial weakly metric association schemes. A connection between this notion and the recent theory of multivariate P-polynomial schemes of ( Bannai et al. 2025) and of m-distance regular graphs ( Bernard et al 2025) is pointed out.

We give upper bounds on the covering radius of special low rate codes (repetition, Simplex, and MacDonald) over ℤ_q , with q a prime power, for the Lee, euclidean and homogeneous distance and compare them to the relevant sphere covering bounds in modest lengths.

In (Solé et al, ISIT 2021), a new notion of bent sequences appeared, where the Sylvester matrix of the Walsh–Hadamard transform is replaced by an arbitrary Hadamard matrix. We introduce an even more general notion of a bent sequence, defined in terms of the following data: a Hadamard matrix of order n defined over the complex q^th roots of unity (a so-called Butson matrix), an algebraic integer in the cyclotomic field of order q, and a Galois automorphism of that field. This new generalization facilitates the existence of self-dual bent sequences in cases where the restricted definition fails. In particular, we construct self-dual bent sequences for various q≤ 60 and lengths n≤ 21. Computational construction methods comprise the resolution of polynomial systems by Gröbner bases and eigenspace computations. Infinite families of self-dual bent sequences can be constructed from regular Hadamard matrices, Bush-type Hadamard matrices, and generalized Boolean bent functions. As an application of Hadamard bent sequences, we derive a lower bound on the covering radius of the ℤ_q -code attached to a Hadamard matrix, for the so-called Chinese Euclidean metric. To every Hadamard matrix we attach a spherical code, which is optimal as a packing code when q=4. This code is a strength 2 spherical design under some mild conditions. Based on the covering properties of spherical designs, we bound above the covering radius of that spherical code for the standard Euclidean metric. This upper bound, in turn, yields an upper bound on the covering radius of the ℤ_q -code for the Chinese Euclidean metric which is not far from the lower bound.

Many authors have generalized Boolean functions with values over finite fields to functions valued over ℤ_p^k . In this work, we generalize their main properties to functions with values in Galois rings. We mainly study relations between generalized bent functions and their classical component functions. A connection with affine spaces of classical bent functions is established.

In this paper, we consider the commutative non-unitary ring of order four defined as I = {a,b|2a = 0, 2b = 0,a(2) = b,ab = 0}. Self-orthogonal codes for a symplectic inner product over I are introduced. A mass formula to enumerate them under symplectic equivalence is given. An application is a classification of such codes in short lengths.

A multisecret-sharing scheme is a method for distributing n randomly associated secrets s1, s2,..., sn among a set of attendants. Linear codes with complementary duals, LCD codes, form an important class of linear codes, while weighing orthogonal matrices play a significant role in cryptography. In this paper, we design novel multisecret-sharing schemes based on LCD codes derived from weighing orthogonal matrices. We analyze the access structures, coalition statistics, security aspects, and information-theoretic efficiency compared to existing multisecret-sharing methods.