The Relation between constants in generic and degenerate subspaces of free unital associative complex algebra

From the study of the constants in the generic and the degenerate weight subspaces of the free unitary associative complex algebra B, it follows that the constants in the degenerate weight subspaces of the algebra B can be constructed from the corresponding constants in the generic case by a certain specialization procedure. Here we consider that each constant in each generic weight subspace of the algebra B can be expressed by certain iterated q-commutators

Left to right maxima in Dyck prefixes

In a Dyck path, a peak which is strictly (weakly) higher than all the preceding peaks is called a strict (weak) left-to-right maximum. By dropping the restrictions for the path to end on the $x$-axis, one obtains Dyck prefixes.We obtain explicit generating functions for both weak and strict left-to-right maxima in Dyck prefixes.The proofs of the associated asymptotics make use of analytic techniques such as Mellin transforms, singularity analysis and formal residue calculus

Width-k Eulerian polynomials of type A and B: the \gamma-positivity

In this paper, we introduce some new generalizations of classical descent and inversion statistics on signed permutations that arise from the work of Sack and Ulfarsson [18], and called k-width descents and k-width inversions of type A [8]. Using the aforementioned new statistics, we derive new generalizations of Eulerian polynomials of type A, B and D. We establish also the $\gamma$-positivity of the Eulerian "width-k" polynomials. Referring to Petersen's paper [16], we give a combinatorial interpretation of finite sequences associated with these new polynomials using quasi-symmetric functions and a partition P

New Results and Bounds on codes over GF(17)

Determining the best possible values of the parameters of a linear code is one of the most fundamental and challenging problems in coding theory. There exist databases of best-known linear codes (BKLC) over small finite fields. In this work, we establish a database of BKLCs over the field GF(17) together with upper bounds on the minimum distances for lengths up to 150 and dimensions up to 6. In the process, we have found many new linear codes over GF(17). &nbsp

Set-independence graphs of vector spaces and partial quasigroups

As a generalization of independence graphs of vector spaces and groups, we introduce the notions of set-independence graphs of vector spaces and partial quasigroups. The former are characterized for finite-dimensional vector spaces over finite fields. Further, we prove that every finite simple graph is isomorphic to either the independence graph of a partial quasigroup or an induced subgraph of the latter. We also prove that isomorphic partial quasigroups give rise to isomorphic set-independence graphs. As an illustrative example, all finite graphs of order $n\leq 5$ are identified with the independence graph of a partial quasigroup of the same order

On the structure of monomial codes and their generalizations

In this paper, we are interested in monomial codes with associated vector $a=(a_0, a_1,\ldots, a_{n-1}),$ introduced in \cite{Maria2017}, and more generally in linear codes invariant under a monomial matrix M=\diag(a_0, a_1,\ldots, a_{n-1}) P_{\sigma} where $\sigma$ is a permutation and $P_{\sigma}$ its associated permutation matrix.We discuss some connections between monomial codes and codes invariant under an arbitrary monomial matrix $M$. Next, we identify monomial codes with associated vector $a=(a_0,a_2,\ldots, a_{n-1})$ by the ideals of the polynomial ring R_{_{q,n}}:= \quot{{\Fq[x]}}{{\langle x^{n}-\prod_{i=0}^{n-1}a_i \rangle}}, via a special isomorphism $\varphi_{_{\overline{a}}}$ which preserves the Hamming weight and differs from the classical isomorphism used in the case of cyclic codes and their generalizations. This correspondence leads to some basic characterizations of monomial codes such as generator polynomials, parity check polynomials, and others. Next, we focus on the structure of $\ell-$quasi-monomial ( $\ell-$QM) codes of length $n=m\ell,$ where on the one hand, we characterize them by the $R_{_{q,m}}-$submodules of $R_{_{q,m}}^{\ell}.$ On the other hand, $\ell-$QM codes are seen as additive monomial codes over the extension \mathbb{F}_{q^{\ell}}/\Fq. So, as in the case of quasi-cyclic codes \cite{Guneri2018}, we characterize those codes that have $\mathbb{F}_{q^{\ell}}-$linear images with respect to a basis of the extension \mathbb{F}_{q^{\ell}}/\Fq, based on the CRT decomposition. Finally, we show that $\ell-$QM codes and additive monomial codes are asymptotically good

Positive harmonic functions on biregular trees

We show that if f is a positive harmonic function on a biregular tree which has maximal growth along an infinite path in the tree, then every harmonic function g on the tree with 0 ≤ g ≤ f is a multiple of f, thus generalizing a result of Cartier about regular trees

A class of permutation polynomials over the group ring F_pC_p^n

Let p be a prime number and F_pC_p^n denote the group ring of a cyclic group of order pn over Fp. We study the permutation property of the polynomial a_0 + a_1x + a_px^p + · · · + a_p^n x^(p^n) with coefficients a_i ∈ F_pC_p^n where i = 0, 1, p, . . . , p^n. Necessary and sufficient conditions on the coefficients have been obtained so that it becomes a permutation polynomial

Characterization of Totally Real Subfields of 2-Power Cyclotomic Fields and Applications to Signal Set Design

A classification of all totally real subfields K of cyclotomic field Q(zeta_{2^r}), for any r ≥ 4, and the fully-diverse related versions of the Z^n-lattice are presented along with closed-form expressions for their minimum product distance. Any totally real subfield K of Q(zeta_{2^r}) must be of the form K=Q(zeta_{2^2} + zeta_{2^2}^{-1}), where s = r − j for some 0 ≤ j ≤ r − 3. Signal constellations for transmitting information over both Gaussian and Rayleigh fading channels (which can be useful for mobile communications) can be carved out of those lattices

On the parameters of a class of narrow sense primitive BCH codes

The last decades, mark an accelerated progression in the determination of the parameters of the primitive BCH codes. Indeed, BCH codes are powerful in terms of decoding. They are applied in several fields such as: satellite communications, cryptography, compact disk drives... and have good structural properties. Nevertheless, the dimension and the minimum distance of those codes aren't known, in general. In this paper, we present a class of narrow sense primitive BCH codes of designed distance $\delta_{_4}=(q-1)q^{^{m-1}}-1-q^{\lfloor \frac{m+3}{2 }\rfloor}.$ Also, we investigate their Bose distance and the dimension