Gauss sums over some matrix groups

Li, Yan; Hu, Shengmin

doi:10.1016/j.jnt.2012.06.010

Cited by 9 publications

(14 citation statements)

References 19 publications

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“…Recently, we obtained explicit expressions of S(GL n (F q ), U) and S(SL n (F q ), U). (See [12] Theorems 2.1 and 2.4). Such expressions only involve Gauss sums and Kloosterman sums.…”

Section: Preliminariesmentioning

confidence: 97%

On a uniformly distributed phenomenon in matrix groups

Hu¹,

Li²

2013

Journal of Number Theory

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“…Recently, we obtained explicit expressions of S(GL n (F q ), U) and S(SL n (F q ), U). (See [12] Theorems 2.1 and 2.4). Such expressions only involve Gauss sums and Kloosterman sums.…”

Section: Preliminariesmentioning

confidence: 97%

On a uniformly distributed phenomenon in matrix groups

Hu¹,

Li²

2013

Journal of Number Theory

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“…Note that this result should be compared with results of Ferguson et al, Li and Hu, and Karabulutin in the paragraph subsequent to Problem 1.3. In our result, we impose a stronger condition on E, F , i.e., E ⊆ D i and E ⊆ D j , than they did in [15,31,5] , while our threshold q 4 is much better than those in their results (for n = 2).…”

Section: Introductionmentioning

confidence: 64%

“…Ferguson et al [15] developed a version of the Kloosterman sum over matrix rings to prove that if |E||F | ≥ 2q 2n 2 −2 , then there exist elements x ∈ E and y ∈ F such that det(x + y) = 1. In the paper [31], Li and Hu gave an explicit expression of Gauss sum for the special linear group SL n (F q ), and as a consequence, they obtained an improvement of Ferguson et al's result. More precisely, they showed that if n = 2, then the condition |E||F | ≥ Cq 5 is enough, but in higher dimensional cases, we need |E||F | ≥ Cq 2n 2 −2n .…”

Section: Introductionmentioning

confidence: 93%

Distribution of determinant of sum of matrices

Cheong¹,

Koh²,

Pham³

et al. 2019

Preprint

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Let Fq be an arbitrary finite field of order q. In this article, we study det S for certain types of subsets S in the ring M2(Fq) of 2 × 2 matrices with entries in Fq. For i ∈ Fq, let Di be the subset of M2(Fq) defined by Di := {x ∈ M2(Fq) : det(x) = i}. Then our results can be stated as follows. First of all, we show that when E and F are subsets of Di and Dj for some i, j ∈ F * q , respectively, we have det(E + F ) = Fq,

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“…The usual way of computing n q (k × k, k, k) involves the Bruhat decomposition of GL k (F q ) and the theory of q-analogues (see [24], Proposition 1.10.15). A different approach proposed in [16] is based on Gauss sums over finite fields and properties of the Borel subgroup of GL k (F q ). In [2] Buckheister derived a recursive description for n q (k×k, r, k) using an elementary argument, and in [1] Bender applied the results of [2] to provide a closed formula for n q (k×k, r, k).…”

Section: Matrices With Given Rank and H-tracementioning

confidence: 99%

Rank-metric codes and their duality theory

Ravagnani

2015

Des. Codes Cryptogr.

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We compare the two duality theories of rank-metric codes proposed by Delsarte and Gabidulin, proving that the former generalizes the latter. We also give an elementary proof of MacWilliams identities for the general case of Delsarte rank-metric codes. The identities which we derive are very easy to handle, and allow us to re-establish in a very concise way the main results of the theory of rank-metric codes first proved by Delsarte employing the theory of association schemes and regular semilattices. We also show that our identities imply as a corollary the original MacWilliams identities established by Delsarte. We describe how the minimum and maximum rank of a rank-metric code relate to the minimum and maximum rank of the dual code, giving some bounds and characterizing the codes attaining them. Then we study optimal anticodes in the rank metric, describing them in terms of optimal codes (namely, MRD codes). In particular, we prove that the dual of an optimal anticode is an optimal anticode. Finally, as an application of our results to a classical problem in enumerative combinatorics, we derive both a recursive and an explicit formula for the number of k × m matrices over a finite field with given rank and h-trace. * 1 special case of Delsarte codes. It is however not clear in general how the duality theories of these two families of codes relate to each other. This is one of the questions that we address in this work.Both linear Delsarte and Gabidulin codes have interesting applications in information theory. Recently it was shown how to employ them for error correction in non-coherent linear network coding and in coherent linear network coding under an adversarial channel model (see e.g. [21] and the references within). Rank-metric codes were also proposed to secure a network coding communication system against an eavesdropper in a universal way (see [22] for details).Motivated by these applications, in this paper we study the duality theories of linear Delsarte and Gabidulin codes, mainly focusing on their MacWilliams identities. In coding theory, a MacWilliams identity establishes a relation between metric properties of a code and metric properties of the dual code. MacWilliams identities exist for several types of codes and metrics. As Gluesing-Luerssen observed in [12], association schemes provide the most general approach to MacWilliams identities, and apply to both linear and non-linear codes (see [4], [3] and [7]). On the other hand, the machinery of association schemes and of the related Bose-Mesner algebras is a very elaborated mathematical tool. Several authors proved independently the MacWilliams identities for the various types of codes in less sophisticated ways.A different viewpoint on MacWilliams identities for general additive codes was recently proposed by Gluesing-Luerssen in [12]. The approach is based on character theory and partitions of groups. See also [14] for a character-theoretic approach to MacWilliams identities for the rank and the Hamming metric.Both the theory of association sch...

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Gauss sums over some matrix groups

Cited by 9 publications

References 19 publications

On a uniformly distributed phenomenon in matrix groups

On a uniformly distributed phenomenon in matrix groups

Distribution of determinant of sum of matrices

Rank-metric codes and their duality theory

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