Shi et al. (2021) and Cheng (2024) constructed LCD double Toeplitz codes from tridiagonal symmetric and skew-symmetric Toeplitz matrices by factorizing Dickson polynomials. This paper extends their work to linear codes generated by anti-tridiagonal Hankel matrices with zero on the sub-diagonal, denoted H2n+1(b, 0, c). Using a permutation involution, it is shown that H2n+1(b, 0, c) is similar to a symmetric tridiagonal 2-Toeplitz matrix with zero main diagonal, allowing the factorization of the Hankel matrix’s characteristic polynomial via Dickson polynomials. Based on this factorization, necessary and sufficient conditions for the code C2n+1(b, 0, c) = [I2n+1,H2n+1(b, 0, c)] to be LCD are derived over finite fields of both even and odd characteristic. For even characteristic, the LCD condition is expressed in terms of roots of unity and parameters b, c, including an extension when gcd(p, n+1) ̸= 1. For odd characteristic, conditions involve primitive 2(n+1)-th roots of unity and the element μ with μ2 = −1. The results are further generalized to the case pr ∥ (n + 1) by reducing to primitive (m + 1)-th or 2(m + 1)-th roots. These constructions provide new families of LCD codes from Hankel matrices, complementing existing Toeplitz-based constructions and offering flexible parameter choices for side-channel and fault-injection attack resistant cryptography. An example over F4 is given.

Let R m be endowed with the Euclidean metric. The covering radius of a lattice Λ ⊂ R m is the least distance r such that, given any point of R m , the distance from that point to Λ is not more than r . Lattices can occur via the unit group of the ring of integers in an algebraic number field K , by applying a logarithmic embedding K ⁎ → R m . In this paper, we examine those lattices which arise from the cyclotomic number field Q ( ζ n ) , for a given positive integer n ≥ 5 such that n ≢ 2 ( mod 4 ) . We then provide improvements to a result of de Araujo in [3] , and conclude with an upper bound on the covering radius for this lattice in terms of n and the number of its distinct prime factors. In particular, we improve [3, Lemma 2] , and show that, asymptotically, it can be improved no further.

Abstract Let 𝔽 q be the finite field of order q and F = 𝔽 q ( x ) the rational function field. In this paper, we give a characterization of the cyclotomic function fields F ( Λ M ) with modulus M , where M ∈ 𝔽 q [ T ] is a monic and irreducible polynomial of degree two. More precisely, we show that F ( Λ M ) is the only function field, up to 𝔽 q -isomorphism, with q + 1 𝔽 q -rational places, genus ( q + 1)( q − 2)/2 and a subgroup of automorphisms over 𝔽 q isomorphic to F q 2 ∗ . $\mathbb{F}_{q^2}^*.$ We also provide the full automorphism group of F ( Λ M ) in odd characteristic, extending results of [14] where the automorphism group of F ( Λ M ) over 𝔽 q was computed.

Jiang and Rallis (1997) defined a family of local integrals attached to a cubic polynomial and proved explicit evaluations of them over a non-archimedean local field F, when either F contains three third roots of unity, or the defining polynomial is reducible. The restriction on F allowed them, among other things, to reduce the case of irreducible polynomials of the form x3 - a. Pleso (2009) began the work of removing the restriction on F by expressing the integral as a sum of 16 integrals for the cubic polynomial x3 -bx -c, with b,c in F, and computing nine of them. In this work, we compute 15 of Pleso's integrals explicitly, and reduce the last to a conjecture about the number of points on a surface over a finite field, in the special case when F is the p-adic numbers (F = Qp) and p is equivalent to 5 mod 6. The proof of this conjecture is provided in the appendix section. Our computations essentially complete Pleso's work in that special case. In the interim, Xiong (2020) has computed the integrals for an arbitrary non-archimedean local field by a totally different approach. Our direct approach might be more extendable to analogous integrals defined using quintic polynomials in a higher-rank setting.

In this article we classify all simple modules over a noncommutative and noncocommutative bialgebra M ( p , q ) assuming q is a root of unity.

Hermitian adjacency matrices with at most three distinct eigenvalues

Abstract We consider a 2-homogeneous bipartite distance-regular graph $$\Gamma $$ Γ with diameter $$D \ge 3$$ D ≥ 3 . We assume that $$\Gamma $$ Γ is not a hypercube nor a cycle. We fix a Q -polynomial ordering of the primitive idempotents of $$\Gamma $$ Γ . This Q -polynomial ordering is described using a nonzero parameter $$q \in \mathbb {C}$$ q ∈ C that is not a root of unity. We investigate $$\Gamma $$ Γ using an $$S_3$$ S 3 -symmetric approach. In this approach one considers $$V^{\otimes 3} = V \otimes V \otimes V$$ V ⊗ 3 = V ⊗ V ⊗ V where V is the standard module of $$\Gamma $$ Γ . We construct a subspace $$\Lambda $$ Λ of $$V^{\otimes 3}$$ V ⊗ 3 that has dimension $$\left( {\begin{array}{c}D+3\\ 3\end{array}}\right) $$ D + 3 3 , together with six linear maps from $$\Lambda $$ Λ to $$\Lambda $$ Λ . Using these maps we turn $$\Lambda $$ Λ into an irreducible module for the nonstandard quantum group $$U^\prime _q(\mathfrak {so}_6)$$ U q ′ ( so 6 ) introduced by Gavrilik and Klimyk in 1991.

Abstract The incompatibility of measurements is the key feature of quantum theory that distinguishes it from the classical description of nature. Here, we consider groups of d -outcome quantum observables with prime d represented by non-Hermitian unitary operators whose eigenvalues are d th roots of unity. We additionally assume that these observables mutually commute up to a scalar factor being one of the d ’th roots of unity. By representing commutation relations of these observables via a frustration graph, we show that for such a group, there exists a single unitary transforming them into a tensor product of generalized Pauli matrices and some ancillary mutually commuting operators. Building on this result, we derive upper bounds on the sum of the squares of the absolute values and the sum of the expected values of the observables forming a group. Such bounds are of particular importance to deriving uncertainty relations or constructing entanglement witnesses, and are also useful in inflation technique. We finally utilize these bounds to compute the generalized geometric measure of entanglement for qudit stabilizer subspaces.

Physical-layer security (PLS) provides an information-theoretic framework for securing wireless communications by exploiting channel and signal-structure asymmetries, thereby avoiding reliance on computational hardness assumptions. Within this setting, lattice codes and their algebraic constructions play a central role in achieving secrecy over Gaussian and fading wiretap channels. This article offers a comprehensive survey of lattice-based wiretap coding, covering foundational concepts in algebraic number theory, Construction A over number fields, and the structure of modular and unimodular lattice families. We review key secrecy metrics, including secrecy gain, flatness factor, and equivocation, and consolidate classical and recent results to provide a unified perspective that links wireless-channel models with their underlying algebraic lattice structures. In addition, we review a newly proposed family of p-modular lattices in Khodaiemehr, H., 2018 constructed from cyclotomic fields Q(ζp) for primes p≡1(mod4) via a generalized Construction A framework. We characterize their algebraic and geometric properties and establish a non-existence theorem showing that such constructions cannot be extended to prime-power cyclotomic fields Q(ζpn) with n>1. Finally, motivated by the fact that these p-modular lattices naturally yield mixed-signature structures for which classical theta series diverge, we integrate recent advances on indefinite theta series and modular completions. Drawing on Vignéras' differential framework and generalized error functions, we outline how modularly completed indefinite theta series provide a principled analytic foundation for defining secrecy-relevant quantities in the indefinite setting. Overall, this work serves both as a survey of algebraic lattice techniques for PLS and as a source of new design insights for secure wireless communication systems.

A bstract High-temperature ( q → 1) asymptotics of 4d superconformal indices of Lagrangian theories have been recently analyzed up to exponentially suppressed corrections. Here we use RG-inspired tools to extend the analysis to the exponentially suppressed terms in the context of Schur indices of $$ \mathcal{N}=2 $$ N = 2 SCFTs. In particular, our approach explains the curious patterns of logarithms (polynomials in 1 / log q ) found by Dedushenko and Fluder in their numerical study of the high-temperature expansion of rank-1 theories. We also demonstrate compatibility of our results with the conjecture of Beem and Rastelli that Schur indices satisfy finite-order, possibly twisted, modular linear differential equations (MLDEs), and discuss the interplay between our approach and the MLDE approach to the high-temperature expansion. The expansions for q near roots of unity are also treated. A byproduct of our analysis is a proof (for Lagrangian theories) of rationality of the conformal dimensions of all characters of the associated VOA, that mix with the Schur index under modular transformations.

We show that the Schur polynomials in all primitive n th roots of unity are 1, 0, or −1, if n has at most two distinct odd prime factors. This result can be regarded as a generalization of properties of the coefficients of the cyclotomic polynomial and its multiplicative inverse. The key to the proof is the concept of a unimodular system of vectors. Namely, this result can be reduced to the unimodularity of the tensor product of two maximal circuits (here we call a vector system a maximal circuit, if it can be expressed as B ∪ { − ∑ B } with some basis B ).

Abstract We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally real algebraic fields whose rings of integers have finite Pythagoras numbers, namely, one, two, three, and at least four.

Abstract Let G be a finite abelian group. We prove that there exists a positive density subfamily of real cyclotomic fields whose ideal class groups contain a subgroup isomorphic to G .

We give a new proof for the description of the blocks in the category of representations of a reductive algebraic group G over a field of positive characteristic ℓ (originally due to Donkin), by working in the Satake category of the Langlands dual group and applying Smith–Treumann theory as developed by Riche and Williamson. On the representation theoretic side, our methods enable us to give a bound for the length of a minimum chain linking two weights in the same block, and to give a new proof for the block decomposition of a quantum group at an ℓ th root of unity.

We study the Gauss and Jacobi sums from a viewpoint of motives. We exhibit isomorphisms between Chow motives arising from the Artin–Schreier curve and the Fermat varieties over a finite field, that can be regarded as (and yield a new proof of) classically known relations among Gauss and Jacobi sums such as Davenport–Hasse’s multiplication formula. As a key step, we define motivic analogues of the Gauss and Jacobi sums as algebraic correspondences, and show that they represent the Frobenius endomorphisms of such motives. This generalizes Coleman’s result for curves. These results are applied to investigate the group of invertible Chow motives with coefficients in a cyclotomic field.

In this article we study the endomorphism algebras of abelian varieties A defined over a given number field K with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of A to be defined over K(A[2]), the field extension generated by its 2-torsion. When K= mathbb {Q} and textrm{Gal}(mathbb {Q}(A[2])/mathbb {Q}) is cyclic of prime order p = 2 dim (A) +1, we prove that there are only finitely many possibilities for the geometric endomorphism algebra textrm{End}(A) otimes mathbb {Q}. In fact, when dim (A) not in {3,5,9,21,33,81}, we show textrm{End}(A) otimes mathbb {Q} is a proper subfield of the p-th cyclotomic field. In particular, when g=2, textrm{End}(A) otimes mathbb {Q} is isomorphic to either mathbb {Q} or mathbb {Q}(sqrt{5}).

In this article, we investigate the representation of the quantized enveloping algebra U q + ( B 2 ) of the positive nilpotent part of a Lie algebra of type B 2 in the case where the deformation parameter q is a root of unity. We show that the algebra U q + ( B 2 ) is a polynomial identity (PI) algebra at roots of unity. We compute the PI degree of U q + ( B 2 ) and present a classification of its simple modules up to isomorphism, over an algebraically closed field of arbitrary characteristic. In addition, we determine the center of U q + ( B 2 ) .

Using a parent Hermitian tight-binding model on a bipartite lattice with chiral symmetry, we theoretically generate non-Hermitian models for free fermions with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>p</a:mi> </a:math> orbitals per unit cell satisfying a complex generalization of chiral symmetry. The <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>p</b:mi> </b:math> complex energy bands in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>k</c:mi> </c:math> space are given by a common <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>k</d:mi> </d:math> -dependent real factor, determined by the bands of the parent model, multiplied by the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>p</e:mi> <e:mspace width="0.16em"/> <e:mi>th</e:mi> </e:mrow> </e:math> roots of unity. When the parent model is the Su-Schrieffer-Heeger (SSH) model, the single-particle energy levels are the same as those of free parafermion solutions to Baxter's non-Hermitian clock model. This construction relies on fully unidirectional hopping to create Bloch Hamiltonians with the form of generalized permutation matrices, but we also describe the effect of partial unidirectional hopping. For fully bidirectional hopping, the Bloch Hamiltonians are Hermitian and may be separated into even and odd parity blocks with respect to inversion of the orbitals within the unit cell. Partially unidirectional hopping breaks the inversion symmetry and mixes the even and odd blocks, and the real energy spectrum evolves into a complex one as the degree of unidirectionality increases, with details determined by the topology of the parent model and the number of orbitals per unit cell, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mi>p</g:mi> </g:math> . We describe this process in detail for <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mrow> <h:mi>p</h:mi> <h:mo>=</h:mo> <h:mn>3</h:mn> </h:mrow> </h:math> and 4 with the SSH model. We also apply our approach to graphene, and we show that <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mi>A</i:mi> <i:mi>A</i:mi> </i:mrow> </i:math> -stacked bilayer graphene evolves into a square-root Hamiltonian of monolayer graphene with the introduction of unidirectional hopping. We show that higher-order exceptional points occur at edge states and solitons in the non-Hermitian SSH model, and at the Dirac point of non-Hermitian graphene.

Let χ be a primitive Dirichlet character modulo q , and let δ &gt; 0 . Assuming that χ has large order d , for any d th root of unity α we obtain non-trivial upper bounds for the number of n ≤ x such that χ ( n ) = α , provided x &gt; q δ . This improves upon a previous result of the first author by removing restrictions on q and d . As a corollary, we deduce that if the largest prime factor of d satisfies P + ( d ) → ∞ then the level set χ ( n ) = α has o ( x ) such solutions whenever x &gt; q δ , for any fixed δ &gt; 0 . Our proof relies, among other things, on a refinement of a mean-squared estimate for short sums of the characters χ ℓ , averaged over 1 ≤ ℓ ≤ d - 1 , due to the first author, which goes beyond Burgess’ theorem as soon as d is sufficiently large. We in fact show the alternative result that the partial sum of either (a) χ itself, or (b) χ ℓ , for “almost all” 1 ≤ ℓ ≤ d - 1 , exhibits cancellation on the interval [ 1 , q δ ] , for any fixed δ &gt; 0 . By an analogous method, we also show that the Pólya–Vinogradov inequality may be improved for either χ itself or for almost all χ ℓ , with 1 ≤ ℓ ≤ d - 1 . In particular, our averaged estimates are non-trivial whenever χ has sufficiently large even order d .

In this Article the author hopes to present an unusual view of the Riemann zeta landscape when Re ( s ) &gt; 0, to show how the prime numbers are discoverable by the altered configuration, and to show a correlation amongst the primes, the regular polygons and a set of complex numbers with real part ${1 \over 2}$ .