Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of ...commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most
|
N
μ
:
(
N
μ
)
2
N
λ
|
classes with respect to the strong isotopy, where
N
μ
and
N
λ
are the middle and the left nucleus of Q. Loops Q
1
and Q
2
are said to be strongly isotopic if there exists an isotopism
Q
1
→
Q
2
of the form
(
α
,
α
,
γ
)
.
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2.
Generalized twisted Gabidulin codes Lunardon, Guglielmo; Trombetti, Rocco; Zhou, Yue
Journal of combinatorial theory. Series A,
October 2018, 2018-10-00, Volume:
159
Journal Article
Peer reviewed
Open access
Let C be a set of m by n matrices over Fq such that the rank of A−B is at least d for all distinct A,B∈C. Suppose that m⩽n. If #C=qn(m−d+1), then C is a maximum rank distance (MRD for short) code. ...Until 2016, there were only two known constructions of MRD codes for arbitrary 1<d<m−1. One was found by Delsarte (1978) 8 and Gabidulin (1985) 10 independently, and it was later generalized by Kshevetskiy and Gabidulin (2005) 16. We often call them (generalized) Gabidulin codes. Another family was recently obtained by Sheekey (2016) 22, and its elements are called twisted Gabidulin codes. In the same paper, Sheekey also proposed a generalization of the twisted Gabidulin codes. However the equivalence problem for it is not considered, whence it is not clear whether there exist new MRD codes in this generalization. We call the members of this putative larger family generalized twisted Gabidulin codes. In this paper, we first compute the Delsarte duals and adjoint codes of them, then we completely determine the equivalence between different generalized twisted Gabidulin codes. In particular, it can be proven that, up to equivalence, generalized Gabidulin codes and twisted Gabidulin codes are both proper subsets of this family.
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In this paper, we consider Leavitt path algebras having coefficients in a
-semifield. Concentrating on the aspect of
-simplicity, we find a set of necessary and sufficient conditions for the
...-simplicity of the Leavitt path algebra
(Γ) of a directed graph Γ over a non-zeroid
-semifield
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En 4 se estableció la siguiente conjetura: Si un presemicuerpo de Figueroa P(K; alfa; beta; A;B) admite un autotopismo de orden un divisor primo p-primitivo de p^n-1, entonces su grupo autotopismo es ...isomórfo a un subgrupo de GL(K) x GL(K). En 5 esta conjetura se resolvió bajo una condición adicional de normalidad.En este artículo, mostramos que la suposición hecha en la hipótesis de la conjetura es necesaria en el sentido de que existe un presemicuerpo de Figueroa, que no admite tal autotopismo, para el cual la conjetura no se cumple.
In 4 was stated the following conjecture: If a Figueroa’s presemifield P(K; alfa; beta; A;B) admits an autotopism of order a p-primitive prime divisor of p^n-1, then its autotopism group is isomorphic to a subgroup of GL(K) x GL(K). In 5 this conjecture was settled under an additional normality condition. In this article, we show that the assumption in the hypothesis of the conjecture is necessary in the sense that there exist a Figueroa’s presemifield, that does not admit such autotopism, for which the conjecture is not met.
We classify the rank two commutative semifields which are 8-dimensional over their center 𝔽
. This is done using computational methods utilizing the connection to linear sets in PG(2,
). We then ...apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over 𝔽
. The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.
In 8 Dempwolff gives a construction of three classes of rank two semifields of order q2n, with q and n odd, using Dembowski–Ostrom polynomials. The question whether these semifields are new, i.e. not ...isotopic to previous constructions, is left as an open problem. In this paper we solve this problem for n>3, in particular we prove that two of these classes, labeled DA and DAB, are new for n>3, whereas presemifields in family DB are isotopic to Generalized Twisted Fields for each n≥3.
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Bent Partitions and LP-Packings Alkan, Sezel; Anbar, Nurdagul; Kalayci, Tekgul ...
IEEE transactions on information theory,
07/2024, Volume:
70, Issue:
7
Journal Article
Peer reviewed
Recently, the concept of (normal) bent partitions, which are partitions of elementary abelian groups having similar properties to spreads, has been introduced by Anbar and Meidl. A large number of ...bent partitions, so-called generalized semifield spreads, can be obtained from semifields with certain properties. A strongly related concept, namely Latin square partial difference set packings (LP-packings) in finite abelian groups, has also been introduced recently by Jedwab and Li. The examples for LP-packings in an elementary abelian group are obtained from spreads. LP-packings yield bent partitions (not only for elementary abelian groups). In this paper, we first point out that conversely, generalized semifield spreads yield LP-packings. As a result, there is a huge amount of LP-packings in elementary abelian groups, other than spreads. With some examples from ternary bent functions, we then show that normal bent partitions and LP-packings are not the same concept. Finally, we extend the lifting procedure from spreads to LP-packings in nonelementary abelian groups to a lifting procedure from some generalized spreads to LP-packings in nonelementary abelian groups and in larger elementary abelian groups. This potentially yields bent partitions other than generalized semifield spreads.
We consider a decision-making problem to find absolute ratings of alternatives that are compared in pairs under multiple criteria, subject to constraints in the form of two-sided bounds on ratios ...between the ratings. Given matrices of pairwise comparisons made according to the criteria, the problem is formulated as the log-Chebyshev approximation of these matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank) to minimize the approximation errors for all matrices simultaneously. We rearrange the approximation problem as a constrained multiobjective optimization problem of finding a vector that determines the approximating consistent matrix. The problem is then represented in the framework of tropical algebra, which deals with the theory and applications of idempotent semirings and provides a formal basis for fuzzy and interval arithmetic. We apply methods and results of tropical optimization to develop a new approach for handling the multiobjective optimization problem according to various principles of optimality. New complete solutions in the sense of the max-ordering, lexicographic ordering and lexicographic max-ordering optimality are obtained, which are given in a compact vector form ready for formal analysis and efficient computation. We present numerical examples of solving multicriteria problems of rating four alternatives from pairwise comparisons to illustrate the technique and compare it with others.
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A new family of commutative semifields with two parameters is presented. Its left and middle nucleus are both determined. Furthermore, we prove that for different pairs of parameters, these ...semifields are not isotopic. It is also shown that, for some special parameters, one semifield in this family can lead to two inequivalent planar functions. Finally, using a similar construction, new APN functions are given.
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We determine the tensor rank of all semifields of order 16 over F2 and of all semifields of order 81 over F3. Our results imply that some semifields of order 81 have lower multiplicative complexity ...than the finite field F81 over F3. We prove new results on the correspondence between linear codes and tensor rank, including a generalisation of a theorem of Brockett and Dobkin to arbitrary tensors, which makes the problem computationally feasible.
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