Algebraic structure of codes over Fq , closed under arbitrary abelian group G of permutations with exponent relatively prime to q, called G-invariant codes, is investigated using a transform domain approach. In particular, this general approach unveils algebraic structure of quasicyclic codes, abelian codes, cyclic codes, and quasi-abelian codes with restriction on G to appropriate special case...

This work is intended to introduce the theories of dual-generalized complex and hyperbolicgeneralized numbers. The algebraic properties these numbers are taken into consideration. Besides, hyperbolic-generalized valued functions defined different matrix representations examined. Moreover, a remarkable classification given for special cases set complexgeneralized mentioned.

Consider a connected topological space $$X$$ with point $$x$$ in and let $$K$$ be field the discrete topology. We study Tannakian category of finite-dimensional (flat) vector bundles on its dual $$\pi(X,x)$$ respect to fiber functor . The maximal pro-étale quotient is étale fundamental group studied by Kucharczyk Scholze. For well-behaved spaces, pro-algebraic completion ordinary group. obtain ...

Given a group $G$ of automorphisms matroid $M$, we describe the representations on homology independence complex dual $M^*$. These are related with lattice flats and (when $M$ is realizable) top cohomology hyperplane arrangement. Finally analyze in detail case complete graph, which has applications to algebraic geometry.

A flag of codes C0 ⊊ C1 ... Cs ⊆ Fnq is said to satisfy the isometry-dual property if there exists x ∈ (F*q)n such that code Ci x-isometric dual C⊥s-i for all i = 0, ..., s. For P and Q rational places in a function field F, we investigate existence flags families two-point algebraic geometry CL(D, a0P + bQ) a1P ...⊊ asP bQ), where divisor D sum pairwise different F P,Q are not supp(D). We char...

We provide a systematic way to design computable bilinear forms which, on the class of subspaces $W^* \subseteq \mathcal{V}'$ that can be obtained by duality from given finite dimensional subspace $W$ an Hilbert space $\mathcal{V}$, are spectrally equivalent scalar product $\mathcal{V}'$. Such form used build stabilized discretization algorithm for solution abstract saddle point problem allowin...

The purpose of this short note is to show that the Christoffel–Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving dual problem semi-algebraic D-optimal experimental design statistics. It uses only elementary notions convex analysis. Geometric interpretations algorithmic consequences are mentioned.

Assaf and McNamara [1] recently used combinatorics to give an elegant and surprising formula for the product of a skew Schur function by a complete homogeneous symmetric function. Their paper included an appendix by one of us (Lam) with a simple algebraic proof of their formula, and also a conjectural skew version of the Littlewood-Richardson rule. We show how these formulas and much more are s...

It was shown by Massey that linear complementary dual (LCD for short) codes are asymptotically good. In 2004, Sendrier proved that LCD codes meet the asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound still remains to be the best asymptotical lower bound for LCD codes. In this paper, we show that an algebraic geometry code over a finite field of even characteristic is eq...

This paper reviews and compares constructive and algebraic approaches in the study of rough sets. In the constructive approach, one starts from a binary relation and defines a pair of lower and upper approximation operators using the binary relation. Different classes of rough set algebras are obtained from different types of binary relations. In the algebraic approach, one defines a pair of du...