Mixing of the upper triangular matrix walk

We study a natural random walk over the upper triangular matrices, with entries in the field ${\mathbb{Z}_2}$ , generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n ²) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields ${\mathbb{Z}_q}$ for q prime.     

A class of constacyclic codes over a finite field-II

Let \(\mathbb{F}_q\) be a finite field with q = p ^m elements, where p is any prime and m ≥ 1. In this paper, we explicitly determine all the μ-constacyclic codes of length ? ⁿ over \(\mathbb{F}_q\), where ? is an odd prime coprime to p and the order of μ is a power of ?. All the repeated-root λ- constacyclic codes of length ? ⁿ p ^s over \(\mathbb{F}_q\) are also determined for any nonzero λ in \(\mathbb{F}_q\). As examples all the λ-constacyclic codes of length 3 ⁿ p ^s over \(\mathbb{F}_q\) for p = 5, 7, 11, 19 for n ≥ 1, s ≥ 1 are derived. We also obtain all the self-orthogonal negacyclic codes of length ? ⁿ over \(\mathbb{F}_q\) when q is odd prime power and give some illustrative examples.     

Degree complexity of birational maps related to matrix inversion: symmetric case

For q?≥ 3, we let ${\mathcal{S}_q}$ denote the projectivization of the set of symmetric q × q matrices with coefficients in ${\mathbb{C}}$ . We let ${I(x)=(x_{i,j})^{-1}}$ denote the matrix inverse, and we let ${J(x)=(x_{i,j}^{-1})}$ be the matrix whose entries are the reciprocals of the entries of x. We let ${K|\mathcal{S}_q=I\circ J:~\mathcal{S}_q\rightarrow \mathcal{S}_q}$ denote the restriction of the composition I ? J to ${\mathcal{S}_q}$ . This is a birational map whose properties have attracted some attention in statistical mechanics. In this paper we compute the degree complexity of ${K|\mathcal{S}_q}$ , thus confirming a conjecture of Angles d’Auriac et?al. (J Phys A Math Gen 39:3641–3654, 2006).     

Transitive A 6-invariant k-arcs in PG(2, q)

For q = p ^r with a prime p ≥ 7 such that ${q \equiv 1}$ or 19 (mod 30), the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A ₆ of degree 6. For a projectivity group ${\Gamma \cong A_6}$ of PG(2, q), we investigate the geometric properties of the (unique) Γ-orbit ${\mathcal{O}}$ of size 90 such that the 1-point stabilizer of Γ in its action on ${\mathcal O}$ is a cyclic group of order 4. Here ${\mathcal O}$ lies either in PG(2, q) or in PG(2, q ²) according as 3 is a square or a non-square element in GF(q). We show that if q ≥ 349 and q ≠ 421, then ${\mathcal O}$ is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601,661. Interestingly, ${\mathcal O}$ is the smallest known complete arc in PG(2,601) and in PG(2,661). Computations are carried out by MAGMA.     

On codes over $$\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}$$

In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring \(R=\mathbb {F}_{q}+v\mathbb {F}_{q}+v^{2}\mathbb {F}_{q}\), where \(v^{3}=v\), for q odd. We give conditions on the existence of LCD codes and present construction of formally self-dual codes over R. Further, we give bounds on the minimum distance of LCD codes over \(\mathbb {F}_q\) and extend these to codes over R.     

A problem of Carlitz and its generalizations

Let ${\mathbb{F}_q}$ be the finite field of characteristic p > 2 with q elements. Carlitz proposed the problem of finding an explicit formula for the number of solutions to the equation $$(x_1+ x_2+\cdots+x_n)^2=a\, x_1x_2\cdots x_n,$$ where ${a\in \mathbb{F}_q^*}$ and n ≥ 3. By using the augmented degree matrix and Gauss sums, we consider the generalizations of the above equation and partially solve Carlitz’s problem. Moreover, the technique developed in this paper may be applied to other equations of the form ${h_1^\lambda=h_2}$ with ${h_1, h_2 \in \mathbb{F}_q[x_1,\ldots,x_n]}$ and ${\lambda \in \mathbb{N}}$ .     

On the reducibility of some composite polynomials over finite fields

Let g(x)?=?x ⁿ ?+?a _n-1 x ^n-1?+?. . .?+?a ₀ be an irreducible polynomial over ${\mathbb{F}_q}$ . Varshamov proved that for a?=?1 the composite polynomial g(x ^p ?ax?b) is irreducible over ${\mathbb{F}_q}$ if and only if ${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})\neq 0}$ . In this paper, we explicitly determine the factorization of the composite polynomial for the case a?=?1 and ${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})= 0}$ and for the case a?≠ 0, 1. A recursive construction of irreducible polynomials basing on this composition and a construction with the form ${g(x^{r^kp}-x^{r^k})}$ are also presented. Moreover, Cohen’s method of composing irreducible polynomials and linear fractions are considered, and we show a large number of irreducible polynomials can be obtained from a given irreducible polynomial of degree n provided that gcd(n, q ³ ? q)?=?1.     

On {\mathbb{F}_q} -Rational Structure of Nilpotent Orbits in the Witt Algebra

Let ${\mathfrak{g}=W_1}$ be the p-dimensional Witt algebra over an algebraically closed field ${k=\overline{\mathbb{F}}_q}$ , where p > 3 is a prime and q is a power of p. Let G be the automorphism group of ${\mathfrak{g}}$ . The Frobenius morphism F _G (resp. ${F_\mathfrak{g}}$ ) can be defined naturally on G (resp. ${\mathfrak{g}}$ ). In this paper, we determine the ${F_\mathfrak{g}}$ -stable G-orbits in ${\mathfrak{g}}$ . Furthermore, the number of ${\mathbb{F}_q}$ -rational points in each ${F_\mathfrak{g}}$ -stable orbit is precisely given. Consequently, we obtain the number of ${\mathbb{F}_q}$ -rational points in the nilpotent variety.     

On zeros of Kloosterman sums

A Kloosterman zero is a non-zero element of ${{\mathbb F}_q}$ for which the Kloosterman sum on ${{\mathbb F}_q}$ attains the value 0. Kloosterman zeros can be used to construct monomial hyperbent (bent) functions in even (odd) characteristic, respectively. We give an elementary proof of the fact that for characteristic 2 and 3, no Kloosterman zero in ${{\mathbb F}_q}$ belongs to a proper subfield of ${{\mathbb F}_q}$ with one exception that occurs at q = 16. It was recently proved that no Kloosterman zero exists in a field of characteristic greater than 3. We also characterize those binary Kloosterman sums that are divisible by 16 as well as those ternary Kloosterman sums that are divisible by 9. Hence we provide necessary conditions that Kloosterman zeros must satisfy.     

On self-dual negacirculant codes of index two and four

We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring \(R=\mathbb {F}_q[x]/\langle x^n+1\rangle , \) when they are self-dual and of length 2n or 4n. In particular, in order for the decomposition to be amenable to analysis, we study factorizations of \(x^n+1\) over \(\mathbb {F}_q, \) with n twice an odd prime, containing only three irreducible factors, all self-reciprocal. We give arithmetic conditions bearing on n and q for this to happen. Given a fixed q,  we show these conditions are met for infinitely many n’s, provided a refinement of Artin primitive root conjecture holds. This number theory conjecture is known to hold under generalized Riemann hypothesis (GRH). We derive a modified Varshamov–Gilbert bound on the relative distance of the codes considered, building on exact enumeration results for given n and q.     

Small complete arcs in Galois planes

In this paper we construct a class of k-arcs in PG(2, q), q=p ^h, h>1, p≠3 and prove its completeness for h large enough. The main result states that this class contains complete k-arcs with $$k \leqslant 2 \cdot q^{{9 \mathord{\left/ {\vphantom {9 {10}}} \right. \kern-\nulldelimiterspace} {10}}} {\text{ }}\left( {10{\text{ divides }}h{\text{ and }}q{\text{ }} \geqslant {\text{ }}q_{\text{0}} } \right).$$ Such complete k-arcs are the unique known complete k-arcs with $$k \leqslant {q \mathord{\left/ {\vphantom {q 4}} \right. \kern-\nulldelimiterspace} 4}.$$      

Point configurations on the projective line over a finite field

We study n-point configurations in \({\mathbb{P}^1(\mathbb{F}_q)}\) modulo projective equivalence. For n = 4 and 5, a complete classification is given, along with the numbers of such configurations with a given symmetry group. Using Polya’s coloring theorem, we investigate the behavior of the numbers C(n, q) of classes of n-configurations resp. C ^spec(n, q) of classes with nontrivial symmetry group. Both are described by rational polynomials in q which depend on q modulo \({\lambda(n) = {\rm lcm} \{m \in \mathbb{N} | m \leq n\}}\) .     

Classification and nonexistence results for linear codes with prescribed minimum distances

Starting from a linear [n, k, d] _q code with dual distance ${d^{\bot}}$ , we may construct an ${[n - d^\bot, k - d^\bot +1,\geq d]_q}$ code with dual distance at least ${\left\lceil\frac{d^\bot}{q}\right\rceil}$ using construction Y ₁. The inverse construction gives a rule for the classification of all [n, k, d] _q codes with dual distance ${d^{\bot}}$ by adding ${d^\bot}$ further columns to the parity check matrices of the smaller codes. Isomorph rejection is applied to guarantee a small search space for this iterative approach. Performing a complete search based on this observation, we are able to prove the nonexistence of linear codes for 16 open parameter sets [n, k, d] _q , q =  2, 3, 4, 5, 7, 8. These results imply 217 new upper bounds in the known tables for the minimum distance of linear codes and establish the exact value in 109 cases.     

Composed products and factors of cyclotomic polynomials over finite fields

Let q = p ^s be a power of a prime number p and let ${\mathbb {F}_{\rm q}}$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over ${\mathbb {F}_{\rm q}}$ . In particular we obtain the explicit factorization of the cyclotomic polynomial ${\Phi_{2^nr}}$ over ${\mathbb {F}_{\rm q}}$ where both r ≥ 3 and q are odd, gcd(q, r) = 1, and ${n\in \mathbb{N}}$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of ${\Phi_r}$ over ${\mathbb {F}_{\rm q}}$ is given to us as a known. Let ${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$ be the factorization of ${n \in \mathbb{N}}$ into powers of distinct primes p _i , 1 ≤ i ≤ s. In the case that the multiplicative orders of q modulo all these prime powers ${p_i^{e_i}}$ are pairwise coprime, we show how to obtain the explicit factors of ${\Phi_{n}}$ from the factors of each ${\Phi_{p_i^{e_i}}}$ . We also demonstrate how to obtain the factorization of ${\Phi_{mn}}$ from the factorization of ${\Phi_n}$ when q is a primitive root modulo m and ${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$ Here ${\phi}$ is the Euler’s totient function, and ord _n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over ${\mathbb {F}_{\rm q}}$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334–336, 1984).     

An analog of the weyl decomposition of the spaceL q (ω, ℝ n ) for a first-order differential operatorfor a first-order differential operator

We prove the following theorem: Suppose the function f(x) belongs toL _q (ω, ? ⁿ ), ω ? ? ^m , q∈(1, ∞), and satisfies the inequality $$|\int\limits_\omega {(f(x),{\mathbf{ }}v(x)){\mathbf{ }}dx| \leqslant \mu ||} v||'_q ,{\mathbf{ }}\tfrac{1}{q} + \tfrac{1}{{q'}} = 1,$$ for all n-dimensional vector-valued functions in the kernel of a scalar-valued first-order differential operator £ for which the second-order operatorLL ^* is elliptic. Then there exists a function p(x)∈W _q ¹ (ω) such that $$||f(x) - \mathfrak{L}^* p(x)||q \leqslant C_q \mu .$$ Bibliography: 6 titles.     

Linear and Algebraic Independence of q-Zeta Values

In the paper, results on linear and algebraic independence of q-series of the form $\varsigma _q (s) = \sum\nolimits_{n = 1}^\infty {\sigma _{s - 1} (n)q^n }$ over the field ?(q) are obtained, where $\sigma _{s - 1} (n) = \sum\nolimits_{d|n} {d^{s - 1} }$ , s = 1, 2,... .     

Properties of finite unrefinable chains of ring topologies

Let R(+, ·) be a nilpotent ring and $ \left( {\mathfrak{M}, < } \right) $ be the lattice of all ring topologies on R(+, ·) or the lattice of all such ring topologies on R(+, ·) in each of which the ring R possesses a basis of neighborhoods of zero consisting of subgroups. Let ?? and ??? be ring topologies from $ \mathfrak{M} $ such that $ \tau = {\tau_0}{ \prec_\mathfrak{M}}{\tau_1}{ \prec_\mathfrak{M}} \cdots { \prec_\mathfrak{M}}{\tau_n} = \tau ^{\prime} $ . Then k????n for every chain $ \tau = {\tau ^{\prime}_0} < {\tau ^{\prime}_1} < \cdots < {\tau ^{\prime}_k} = \tau ^{\prime} $ of topologies from $ \mathfrak{M} $ , and also n?=?k if and only if $ {\tau ^{\prime}_i}{ \prec_\mathfrak{M}}{\tau ^{\prime}_{i + 1}} $ for all 0????i?<?k.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

A new multi-linear universal hash family

A new universal hash family is described which generalises a previously known multi-linear hash family. Messages are sequences over a finite field ${\mathbb{F}_q}$ while keys are sequences over an extension field ${\mathbb{F}_{q^n}}$ . A linear map ${\psi}$ from ${\mathbb{F}_{q^n}}$ to itself is used to compute the output digest. Of special interest is the case q = 2. For this case, we show that there is an efficient way to implement ${\psi}$ using a tower field representation of ${\mathbb{F}_{q^n}}$ . From a practical point of view, the focus of our constructions is small hardware and other resource constrained applications. For such platforms, our constructions compare favourably to previous work.