arXiv subjects

Ferruh Özbudak

Publications and source records attributed to Ferruh Özbudak.

17 recordsLinked to original sources

arXiv2026

(Equivariant) Cohomology of Homogeneous Spaces Associated with Composition Algebras

We study homogeneous spaces arising from embeddings of associative composition algebras of dimensions 2 and 4, with particular emphasis on the Hamiltonian quaternion algebra. For inclusions D \hookrightarrow C, we analyze the (equivariant) geometry and topology of the quotients GL(n,C)/GL(n,D). In the quaternionic case, we prove that these spaces are equivariantly diffeomorphic to homogeneous vector bundles over compact symmetric spaces, and hence admit equivariant deformation retractions onto compact homogeneous models. This description determines their homotopy type and reduces the computation of their cohomological invariants to the compact setting. Using this reduction together with Weyl group invariant theory, we compute the rational equivariant cohomology rings with respect to maximal tori and obtain explicit formulas for the Poincaré polynomials of the quaternionic homogeneous spaces. We further extend the analysis to noncompact symmetric spaces associated with Clifford groups. In this setting, we establish equivariant deformation retractions onto compact homogeneous spaces and derive explicit expressions for their Betti numbers.

math.AG↗

arXiv2026

Some structural properties of mixed orthogonal arrays and their irredundancy

Mixed (asymmetric) orthogonal arrays (MOAs) generalize classical orthogonal arrays by allowing columns over different alphabets. However, their study requires very different structural tools than those used for symmetric orthogonal arrays (OAs), since several key features of the symmetric setting are no longer available in the mixed case, including Euclidean duality, a unique global index, and certain classical bounds. In this paper, we establish three structural results for mixed orthogonal arrays. First, we prove a Singleton-type upper bound and obtain a characterization of MDS and almost-MDS mixed orthogonal arrays. Second, we introduce a trace duality for $\mathbb{F}_q$-linear MOAs over $\prod_{i=1}^{s} \mathbb{F}_{q^{n_i}}$ and establish a correspondence with $\mathbb{F}_q$-linear error-block codes that determines the strength of the MOA via the dual distance of the associated error-block code. Finally, we develop a structural theory of irredundant mixed orthogonal arrays (IrMOAs), motivated by their role in the construction of $t$-uniform and absolutely maximally entangled (AME) quantum states. In the extremal case $t=\lfloor s/2\rfloor$, we prove that $\mathbb{F}_q$-linear IrMOAs with minimum index $1$ (yielding AME states of minimal support) are equivalent to $\mathbb{F}_q$-linear error-block MDS codes.

cs.IT↗

arXiv2025

Generalizing the Bierbrauer-Friedman bound for orthogonal arrays

We characterize mixed-level orthogonal arrays in terms of algebraic designs in a special multigraph. We prove a mixed-level analog of the Bierbrauer-Friedman (BF) bound for pure-level orthogonal arrays and show that arrays attaining it are radius-1 completely regular codes (equivalently, intriguing sets, equitable 2-partitions, perfect 2-colorings) in the corresponding multigraph. For the case when the numbers of levels are powers of the same prime number, we characterize, in terms of multispreads, additive mixed-level orthogonal arrays attaining the BF bound. For pure-level orthogonal arrays, we consider versions of the BF bound obtained by replacing the Hamming graph by its polynomial generalization and show that in some cases this gives a new bound. Keywords: orthogonal array, algebraic t-design, completely regular code, equitable partition, intriguing set, Hamming graph, Bierbrauer-Friedman bound, additive codes.

math.CO↗

arXiv2025

Evaluation Codes in Bottleneck Metrics

Analogs of Reed-Solomon codes are introduced within the framework of bottleneck poset metrics. These codes are proven to be maximum distance separable. Furthermore, the results are extended to the setting of Algebraic Geometry codes.

cs.IT↗

arXiv2025

Sequence of Numbers of Linear Codes with Increasing Hull Dimensions

The hull of a linear code $C$ is the intersection of $C$ with its dual code. We present and analyze the number of linear $q$-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given dimension $k$ and length $n\ge 2k$ the number of all $[n,k]_q$ linear codes with hull dimension $l$ decreases as $l$ increases. We also present classification results for binary and ternary linear codes with trivial hulls (LCD and self-orthogonal) for some values of the length $n$ and dimension $k$, comparing the obtained numbers with the number of all linear codes for the given $n$ and $k$.

cs.IT↗

arXiv2024

Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes

Quasi-twisted codes are used here as the classical ingredients in the so-called Construction X for quantum error-control codes. The construction utilizes nearly self-orthogonal codes to design quantum stabilizer codes. We expand the choices of the inner product to also cover the symplectic and trace-symplectic inner products, in addition to the original Hermitian one. A refined lower bound on the minimum distance of the resulting quantum codes is established and illustrated. We report numerous record breaking quantum codes from our randomized search for inclusion in the updated online database.

quant-ph↗

arXiv2024

Griesmer Bound and Constructions of Linear Codes in $b$-Symbol Metric

The $b$-symbol metric is a generalization of the Hamming metric. Linear codes, in the $b$-symbol metric, have been used in the read channel whose outputs consist of $b$ consecutive symbols. The Griesmer bound outperforms the Singleton bound for $\mathbb{F}_q$-linear codes in the Hamming metric, when $q$ is fixed and the length is large enough. This scenario is also applicable in the $b$-symbol metric. Shi, Zhu, and Helleseth recently made a conjecture on cyclic codes in the $b$-symbol metric. In this paper, we present the $b$-symbol Griesmer bound for linear codes by concatenating linear codes and simplex codes. Based on cyclic codes and extended cyclic codes, we propose two families of distance-optimal linear codes with respect to the $b$-symbol Griesmer bound.

cs.IT↗

arXiv2023

Quasi-cyclic perfect codes in Doob graphs and special partitions of Galois rings

The Galois ring GR$(4^Δ)$ is the residue ring $Z_4[x]/(h(x))$, where $h(x)$ is a basic primitive polynomial of degree $Δ$ over $Z_4$. For any odd $Δ$ larger than $1$, we construct a partition of GR$(4^Δ) \backslash \{0\}$ into $6$-subsets of type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets of type $\{c,-c,2c\}$ such that the partition is invariant under the multiplication by a nonzero element of the Teichmuller set in GR$(4^Δ)$ and, if $Δ$ is not a multiple of $3$, under the action of the automorphism group of GR$(4^Δ)$. As a corollary, this implies the existence of quasi-cyclic additive $1$-perfect codes of index $(2^Δ-1)$ in $D((2^Δ-1)(2^Δ-2)/{6}, 2^Δ-1 )$ where $D(m,n)$ is the Doob metric scheme on $Z^{2m+n}$.

math.CO↗

arXiv2022

Constructing MRD codes by switching

MRD codes are maximum codes in the rank-distance metric space on $m$-by-$n$ matrices over the finite field of order $q$. They are diameter perfect and have the cardinality $q^{m(n-d+1)}$ if $m\ge n$. We define switching in MRD codes as replacing special MRD subcodes by other subcodes with the same parameters. We consider constructions of MRD codes admitting such switching, including punctured twisted Gabidulin codes and direct-product codes. Using switching, we construct a huge class of MRD codes whose cardinality grows doubly exponentially in $m$ if the other parameters ($n$, $q$, the code distance) are fixed. Moreover, we construct MRD codes with different affine ranks and aperiodic MRD codes. Keywords: MRD codes, rank distance, bilinear forms graph, switching, diameter perfect codes

cs.IT↗

arXiv2021

LCD Codes from tridiagonal Toeplitz matrice

Double Toeplitz (DT) codes are codes with a generator matrix of the form $(I,T)$ with $T$ a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When $T$ is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal examples of binary and ternary LCD codes from DT codes over extension fields.

cs.IT↗

arXiv2020

Subspace Packings -- Constructions and Bounds

The Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. Kötter and Kschischang showed that codes in Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are $q$-analogs of codes in the Johnson scheme, i.e., constant dimension codes. These codes of the Grassmannian $\mathcal{G}_q(n,k)$ also form a family of $q$-analogs of block designs and they are called subspace designs. In this paper, we examine one of the last families of $q$-analogs of block designs which was not considered before. This family, called subspace packings, is the $q$-analog of packings, and was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A subspace packing $t$-$(n,k,λ)_q$ is a set $\mathcal{S}$ of $k$-subspaces from $\mathcal{G}_q(n,k)$ such that each $t$-subspace of $\mathcal{G}_q(n,t)$ is contained in at most $λ$ elements of $\mathcal{S}$. The goal of this work is to consider the largest size of such subspace packings. We derive a sequence of lower and upper bounds on the maximum size of such packings, analyse these bounds, and identify the important problems for further research in this area.

math.CO↗

arXiv2019

Subspace Packings

The Grassmannian ${\mathcal G}_q(n,k)$ is the set of all $k$-dimensional subspaces of the vector space $\mathbb{F}_q^n$. It is well known that codes in the Grassmannian space can be used for error-correction in random network coding. On the other hand, these codes are $q$-analogs of codes in the Johnson scheme, i.e. constant dimension codes. These codes of the Grassmannian ${\mathcal G}_q(n,k)$ also form a family of $q$-analogs of block designs and they are called \emph{subspace designs}. The application of subspace codes has motivated extensive work on the $q$-analogs of block designs. In this paper, we examine one of the last families of $q$-analogs of block designs which was not considered before. This family called \emph{subspace packings} is the $q$-analog of packings. This family of designs was considered recently for network coding solution for a family of multicast networks called the generalized combination networks. A \emph{subspace packing} $t$-$(n,k,λ)^m_q$ is a set $\mathcal{S}$ of $k$-subspaces from ${\mathcal G}_q(n,k)$ such that each $t$-subspace of ${\mathcal G}_q(n,t)$ is contained in at most $λ$ elements of $\mathcal{S}$. The goal of this work is to consider the largest size of such subspace packings.

cs.IT↗

arXiv2017

New cubic self-dual codes of length 54, 60 and 66

We study the construction of quasi-cyclic self-dual codes, especially of binary cubic ones. We consider the binary quasi-cyclic codes of length 3\ell with the algebraic approach of [9]. In particular, we improve the previous results by constructing 1 new binary [54, 27, 10], 6 new [60, 30, 12] and 50 new [66, 33, 12] cubic self-dual codes. We conjecture that there exist no more binary cubic self-dual codes with length 54, 60 and 66.

math.NT↗

arXiv2017

A new class of three-weight linear codes from weakly regular plateaued functions

Linear codes with few weights have many applications in secret sharing schemes, authentication codes, communication and strongly regular graphs. In this paper, we consider linear codes with three weights in arbitrary characteristic. To do this, we generalize the recent contribution of Mesnager given in [Cryptography and Communications 9(1), 71-84, 2017]. We first present a new class of binary linear codes with three weights from plateaued Boolean functions and their weight distributions. We next introduce the notion of (weakly) regular plateaued functions in odd characteristic $p$ and give concrete examples of these functions. Moreover, we construct a new class of three-weight linear $p$-ary codes from weakly regular plateaued functions and determine their weight distributions. We finally analyse the constructed linear codes for secret sharing schemes.

cs.IT↗

arXiv2017

Additive cyclic codes over finite commutative chain rings

Additive cyclic codes over Galois rings were investigated in previous works. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the previous studies, whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples.

cs.IT↗

arXiv2014

Switchings of semifield multiplications

Let $B(X,Y)$ be a polynomial over $\mathbb{F}_{q^n}$ which defines an $\mathbb{F}_q$-bilinear form on the vector space $\mathbb{F}_{q^n}$, and let $ξ$ be a nonzero element in $\mathbb{F}_{q^n}$. In this paper, we consider for which $B(X,Y)$, the binary operation $xy+B(x,y)ξ$ defines a (pre)semifield multiplication on $\mathbb{F}_{q^n}$. We prove that this question is equivalent to finding $q$-linearized polynomials $L(X)\in\mathbb{F}_{q^n}[X]$ such that $Tr_{q^n/q}(L(x)/x)\neq 0$ for all $x\in\mathbb{F}_{q^n}^*$. For $n\le 4$, we present several families of $L(X)$ and we investigate the derived (pre)semifields. When $q$ equals a prime $p$, we show that if $n>\frac{1}{2}(p-1)(p^2-p+4)$, $L(X)$ must be $a_0 X$ for some $a_0\in\mathbb{F}_{p^n}$ satisfying $Tr_{q^n/q}(a_0)\neq 0$. Finally, we include a natural connection with certain cyclic codes over finite fields, and we apply the Hasse-Weil-Serre bound for algebraic curves to prove several necessary conditions for such kind of $L(X)$.

math.CO↗

arXiv2006

Improved asymptotic bounds for codes using distinguished divisors of global function fields

For a prime power $q$, let $α_q$ be the standard function in the asymptotic theory of codes, that is, $α_q(δ)$ is the largest asymptotic information rate that can be achieved for a given asymptotic relative minimum distance $δ$ of $q$-ary codes. In recent years the Tsfasman-Vlăduţ-Zink lower bound on $α_q(δ)$ was improved by Elkies, Xing, and Niederreiter and Özbudak. In this paper we show further improvements on these bounds by using distinguished divisors of global function fields. We also show improved lower bounds on the corresponding function $α_q^{\rm lin}$ for linear codes.

math.AG↗