Recent zbMATH articles in MSC 15B33https://www.zbmath.org/atom/cc/15B332021-04-16T16:22:00+00:00WerkzeugA new rank metric for convolutional codes.https://www.zbmath.org/1456.150272021-04-16T16:22:00+00:00"Almeida, P."https://www.zbmath.org/authors/?q=ai:almeida.paulo-j"Napp, D."https://www.zbmath.org/authors/?q=ai:napp.diegoSummary: Let \(\mathbb{F}[D]\) be the polynomial ring with entries in a finite field \(\mathbb{F}\). Convolutional codes are submodules of \(\mathbb{F} [D]^n\) that can be described by left prime polynomial matrices. In the last decade there has been a great interest in convolutional codes equipped with a rank metric, called sum rank metric, due to their wide range of applications in reliable linear network coding. However, this metric suits only for delay free networks. In this work we continue this thread of research and introduce a new metric that overcomes this restriction and therefore is suitable to handle more general networks. We study this metric and provide characterizations of the distance properties in terms of the polynomial matrix representations of the convolutional code. Convolutional codes that are optimal with respect to this new metric are investigated and concrete constructions are presented. These codes are the analogs of Maximum Distance Profile convolutional codes in the context of network coding. Moreover, we show that they can be built upon a class of superregular matrices, with entries in an extension field, that preserve their superregularity properties even after multiplication with some matrices with entries in the ground field.Rank functions.https://www.zbmath.org/1456.150012021-04-16T16:22:00+00:00"Beasley, LeRoy B."https://www.zbmath.org/authors/?q=ai:beasley.leroy-bA rank function on an additive monoid \(\mathcal{Q}\) is a function \(f:\mathcal{Q}\rightarrow\mathbb{N}\) such that (i) \(f(A)=0\) if and only if
\(A=0\) and (ii) \(f(A+B)\leq f(A)+f(B)\) for all \(A\) and \(B\). This paper is a short catalogue of examples of rank functions for graphs and for matrices over semirings. For example: for matrices the usual definitions of matrix rank involve rank functions which are all equivalent for matrices over a field but can differ for matrices over a semiring; for any vector space norm \(\left\Vert
~\right\Vert \) over a field \(v\longmapsto\left\lceil \left\Vert v\right\Vert \right\rceil \) is a rank function; and various covering and partition numbers in graphs are rank functions.
For the entire collection see [Zbl 1433.05003].
Reviewer: John D. Dixon (Ottawa)Weil representations via abstract data and Heisenberg groups: a comparison.https://www.zbmath.org/1456.200022021-04-16T16:22:00+00:00"Cruickshank, J."https://www.zbmath.org/authors/?q=ai:cruickshank.james"Gutiérrez Frez, L."https://www.zbmath.org/authors/?q=ai:gutierrez-frez.luis"Szechtman, F."https://www.zbmath.org/authors/?q=ai:szechtman.fernandoThe paper provides Weil representations of unitary groups with even rank over finite rings via Heisenberg groups. The authors use a constructive approach to obtain the explicit matrix form of the Bruhat elements as well as information on generalized Gauss sums. The result is then shown to be identical to the one following from axiomatic considerations. When the ring is local (not necessarily finite) on the other hand, the index of the subgroup generated by the Bruhat elements is computed. Although the subject of the paper is rather technical, all concepts are explained clearly, results are layed down in great detail and proofs are given in a consistent rigorous manner. The authors also provide several examples at the end as well as a nice selection of references. In view of all this, the article might be interesting not only to specialists in the field, but also to graduate students, due to its pedagogical merits.
Reviewer: Danail Brezov (Sofia)Tan's epsilon-determinant and ranks of matrices over semirings.https://www.zbmath.org/1456.150082021-04-16T16:22:00+00:00"Mohindru, Preeti"https://www.zbmath.org/authors/?q=ai:mohindru.preeti"Pereira, Rajesh"https://www.zbmath.org/authors/?q=ai:pereira.rajeshSummary: We use the \(\varepsilon\)-determinant introduced by \textit{Ya-Jia Tan} [Linear Multilinear Algebra 62, No. 4, 498--517 (2014; Zbl 1298.15014)] to define a family of ranks of matrices over certain semirings. We show that these ranks generalize some known rank functions over semirings such as the determinantal rank. We also show that this family of ranks satisfies the rank-sum and Sylvester inequalities. We classify all bijective linear maps which preserve these ranks.Solvability of new constrained quaternion matrix approximation problems based on core-EP inverses.https://www.zbmath.org/1456.150142021-04-16T16:22:00+00:00"Kyrchei, Ivan"https://www.zbmath.org/authors/?q=ai:kyrchei.ivan"Mosić, Dijana"https://www.zbmath.org/authors/?q=ai:mosic.dijana"Stanimirović, Predrag S."https://www.zbmath.org/authors/?q=ai:stanimirovic.predrag-sSummary: Based on the properties of the core-EP inverse and its dual, we investigate three variants of a novel quaternion-matrix (Q-matrix) approximation problem in the Frobenius norm: \(\min \Vert \mathbf{AXB}-\mathbf{C}\Vert_F\) subject to the constraints imposed to the right column space of \(\mathbf{A}\) and the left row space of \(\mathbf{B}\). Unique solution to the considered Q-matrix problem is expressed in terms of the core inverse of \(\mathbf{A}\) and/or the dual core-EP inverse of \(\mathbf{B}\). Thus, we propose and solve problems which generalize a well-known constrained approximation problem for complex matrices with index one to quaternion matrices with arbitrary index. Determinantal representations for solutions of proposed constrained quaternion matrix approximation problems obtained. An example is given to justify obtained theoretical results.