A class of negacyclic BCH codes and its application to quantum codes

In this paper, we study negacyclic BCH codes over \(\mathbb {F}_{q}\) of length \(n=(q^{2m}-1)/(q-1)\), where q is an odd prime power and m is a positive integer. In particular, the dimension, the minimum distance and the weight distribution of some negacyclic BCH codes over \(\mathbb {F}_{q}\) of length \(n=(q^{2m}-1)/(q-1)\) are determined. Two classes of negacyclic BCH codes meeting the Griesmer bound are obtained. As an application, we construct quantum codes with good parameters from this class of negacyclic BCH codes.     

Negacyclic duadic codes

This paper extends the concepts from cyclic duadic codes to negacyclic codes over F_q (q an odd prime power) of oddly even length. Generalizations of defining sets, multipliers, splittings, even-like and odd-like codes are given. Necessary and sufficient conditions are given for the existence of self-dual negacyclic codes over F_q and the existence of splittings of 2N, where N is odd. Other negacyclic codes can be extended by two coordinates in a way to create self-dual codes with familiar parameters.     

Mutually disjoint t-designs and t-SEEDs from extremal doubly-even self-dual codes

It is known that extremal doubly-even self-dual codes of length \(n\equiv 8\) or \(0\ (\mathrm {mod}\ 24)\) yield 3- or 5-designs respectively. In this paper, by using the generator matrices of bordered double circulant doubly-even self-dual codes, we give 3-(n, k; m)-SEEDs with (n, k, m) \(\in \{(32,8,5), (56,12,9), (56,16,9), (56,24,9), (80,16,52)\}\) . With the aid of computer, we obtain 22 generator matrices of bordered double circulant doubly-even self-dual codes of length 48, which enable us to get 506 mutually disjoint 5-(48, k, \(\lambda \) ) designs for (k, \(\lambda \) )=(12, 8),(16, 1356),(20, 36176). Moreover, this implies 5-(48, k; 506)-SEEDs for \(k=12, 16, 20, 24\) .     

Intersections of q-ary perfect codes

The intersections of q-ary perfect codes are under study. We prove that there exist two q-ary perfect codes C ₁ and C ₂ of length N = qn + 1 such that |C ₁ ? C ₂| = k · |P _i |/p for each k ∈ {0,..., p · K ? 2, p · K}, where q = p ^r , p is prime, r ≥ 1, $n = \tfrac{{q^{m - 1} - 1}}{{q - 1}}$ , m ≥ 2, |P _i | = p ^nr(q?2)+n , and K = p ^{n(2r?1)?r(m?1)}. We show also that there exist two q-ary perfect codes of length N which are intersected by p ^nr(q?3)+n codewords.     

Negacyclic self-dual codes over finite chain rings

In this article, we study negacyclic self-dual codes of length n over a finite chain ring R when the characteristic p of the residue field [`(R)]{\bar{R}} and the length n are relatively prime. We give necessary and sufficient conditions for the existence of (nontrivial) negacyclic self-dual codes over a finite chain ring. As an application, we construct negacyclic MDR self-dual codes over GR(p ^t , m) of length p ^m  + 1.     

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.     

Lucas's criterion for the primality of numbers of the form N = h2n−1

The following theorem is proved. Let N = h2ⁿ-1, where n ≥ 2, h is odd, 1 <-h < 2ⁿ, and suppose that v is a positive integer, v ≥ 3,α is a root of the equation $$(v^2 - 4,N) = 1,\left( {\frac{{v - 2}}{N}} \right) = 1,\left( {\frac{{v + 2}}{N}} \right) = - 1$$ . Then for N to be prime, it is necessary and sufficient that s_n?2≡0(modN), where S_k+1=S _k ² ? 2 (k = 0, 1...), s_o=a^h+ a^?h. For given N, an algorithm is described for the construction of the smallest v satisfying the conditions of this theorem.     

Random systems of equations in free abelian groups

We study the solvability of random systems of equations on the free abelian group ? ^m of rank m. Denote by SAT(? ^m , k, n) and \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) the sets of all systems of n equations of k unknowns in ? ^m satisfiable in ? ^m and ? ^m respectively. We prove that the asymptotic density \(\rho \left( {SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)} \right)\) of the set \(SAT_{\mathbb{Q}^m } (\mathbb{Z}^m ,k,n)\) equals 1 for n ≤ k and 0 for n > k. As regards, SAT(? ^m , k, n) for n < k, some new estimates are obtained for the lower and upper asymptotic densities and it is proved that they lie between (Π _j=k?n+1 ^k ζ(j))^?1 and \(\left( {\tfrac{{\zeta (k + m)}} {{\zeta (k)}}} \right)^n\) , where ξ(s) is the Riemann zeta function. For n ≤ k, a connection is established between the asymptotic density of SAT(? ^m , k, n) and the sums of inverse greater divisors over matrices of full rank. Starting from this result, we make a conjecture about the asymptotic density of SAT(? ^m , n, n). We prove that ρ(SAT(? ^m , k, n)) = 0 for n > k.     

Mass formula and structure of self-dual codes over {{\bf Z}_{2^s}}

We look at the structure of and give a mass formula for self-dual codes over the ring ${{\bf Z}_{2^s}}$ of integers modulo 2 ^s . Together with earlier work on the case of odd primes, this completes the mass formula for self-dual codes for ${{\bf Z}_{p^s}}$ , for all prime numbers p and positive integers s.     

On the odd harmonious graphs with applications

The necessary conditions for the existence of odd harmonious labelling of graph are obtained. A cycle C _n is odd harmonious if and only if n≡0 (mod 4). A complete graph K _n is odd harmonious if and only if n=2. A complete k-partite graph K(n ₁,n ₂,…,n _k ) is odd harmonious if and only if k=2. A windmill graph K _n ^t is odd harmonious if and only if n=2. The construction ways of odd harmonious graph are given. We prove that the graph ∨ _i=1 ⁿ G _i , the graph G(+r ₁,+r ₂,…,+r _p ), the graph $\bar{K_{m}}+_{0}P_{n}+_{e}\bar{K_{t}}$ , the graph G∪(X+∪ _k=1 ⁿ Y _k ), some trees and the product graph P _m ×P _n etc. are odd harmonious. The odd harmoniousness of graph can be used to solve undetermined equation.     

On twin and anti-twin words in the support of the free Lie algebra

Let ${\mathcal{L}}_{K}(A)$ be the free Lie algebra on a finite alphabet A over a commutative ring K with unity. For a word u in the free monoid A ^? let $\tilde{u}$ denote its reversal. Two words in A ^? are called twin (resp. anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie polynomial. Let l denote the left-normed Lie bracketing and ?? be its adjoint map with respect to the canonical scalar product on the free associative algebra K??A??. Studying the kernel of ?? and using several techniques from combinatorics on words and the shuffle algebra , we show that, when K is of characteristic zero, two words u and v of common length n that lie in the support of ${\mathcal{L}}_{K}(A)$ ??i.e., they are neither powers a ⁿ of letters a??A with exponent n>1 nor palindromes of even length??are twin (resp. anti-twin) if and only if u=v or $u = \tilde{v}$ and n is odd (resp. $u =\tilde{v}$ and n is even).     

Growth Polynomials for Additive Quadruples and (h,k)-Tuples

Consider the interval of integers I _m,n = {m, m + 1, m + 2, . . . , m + n ? 1}. For fixed integers h, k, m, and c, let \({\Phi^{(c)}_{h,k,m}(n)}\) denote the number of solutions of the equation (a ₁ +  . . . +  a _h ) ? (a _h+1 +  . . . +  a _h+k ) =  c with \({a_i \in I_{m,n}}\) for all i =  1, . . . , h +  k. This is a polynomial in n for all sufficiently large n, and the growth polynomial is constructed explicitly.     

Binomial coefficients, Catalan numbers and Lucas quotients

Let p be an odd prime and let a,m ∈ Z with a 0 and p ︱ m.In this paper we determinep ∑k=0 pa-1(2k k=d)/mk mod p2 for d=0,1;for example,where(-) is the Jacobi symbol and {un}n≥0 is the Lucas sequence given by u0 = 0,u1 = 1 and un+1 =(m-2)un-un-1(n = 1,2,3,...).As an application,we determine ∑0kpa,k≡r(mod p-1) Ck modulo p2 for any integer r,where Ck denotes the Catalan number 2kk /(k + 1).We also pose some related conjectures.     

On the difference between an integer and its m-th power mod n

For any real constants λ ₁, λ ₂ ∈ (0, 1], let $n \geqslant \max \{ [\tfrac{1} {{\lambda _1 }}],[\tfrac{1} {{\lambda _2 }}]\} $ , m ? 2 be integers. Suppose integers a ∈ [1, λ ₁ n] and b ∈ [1, λ ₂ n] satisfy the congruence b ≡ a ^m (mod n). The main purpose of this paper is to study the mean value of (a ? b)^2k for any fixed positive integer k and obtain some sharp asymptotic formulae.     

The existence of maximal (q 2, 2)-arcs in projective Hjelmslev planes over chain rings of length 2 and odd prime characteristic

We prove that (q ², 2)-arcs exist in the projective Hjelmslev plane PHG(2, R) over a chain ring R of length 2, order |R| = q ² and prime characteristic. For odd prime characteristic, our construction solves the maximal arc problem. For characteristic 2, an extension of the above construction yields the lower bound q ² + 2 on the maximum size of a 2-arc in PHG(2, R). Translating the arcs into codes, we get linear [q ³, 6, q ³ ?q ² ?q] codes over ${\mathbb {F}_q}$ for every prime power q > 1 and linear [q ³ + q, 6,q ³ ?q ² ?1] codes over ${\mathbb {F}_q}$ for the special case q = 2 ^r . Furthermore, we construct 2-arcs of size (q + 1)²/4 in the planes PHG(2, R) over Galois rings R of length 2 and odd characteristic p ².     

Cone Dependence—A Basic Combinatorial Concept

We call A ? $ \mathbb{E} $ ⁿ cone independent of B ? $ \mathbb{E} $ ⁿ , the euclidean n-space, if no a = (a ₁,..., a _n ) ∈ A equals a linear combination of B \ {a} with non-negative coefficients. If A is cone independent of A we call A a cone independent set. We begin the analysis of this concept for the sets P(n) = {A ? {0, 1} ⁿ ? $ \mathbb{E} $ ⁿ : A is cone independent} and their maximal cardinalities c(n) ? max{|A| : A ∈ P(n)}. We show that lim _{n → ∞} $ \frac{{c\left( n \right)}}{{2^n }} $ > $\frac{1}{2}$ , but can't decide whether the limit equals 1. Furthermore, for integers 1 < k < ? ≤ n we prove first results about c _n (k, ?) ? max{|A| : A ∈ P _n (k, ?)}, where P _n (k, ?) = {A : A ? V ⁿ _k and V ⁿ _? is cone independent of A} and V ⁿ _k equals the set of binary sequences of length n and Hamming weight k. Finding c _n (k, ?) is in general a very hard problem with relations to finding Turan numbers.     

Optimal conflict-avoiding codes of odd length and weight three

A conflict-avoiding code (CAC) \({\mathcal{C}}\) of length n and weight k is a collection of k-subsets of \({\mathbb{Z}_{n}}\) such that \({\Delta (x) \cap \Delta (y) = \emptyset}\) for any \({x, y \in \mathcal{C}}\) , \({x\neq y}\) , where \({\Delta (x) = \{a - b:\,a, b \in x, a \neq b\}}\) . Let CAC(n, k) denote the class of all CACs of length n and weight k. A CAC with maximum size is called optimal. In this paper, we study the constructions of optimal CACs for the case when n is odd and k = 3.     

Minimal quadratic residue cyclic codes of length 2 n

LetF be a finite field of prime power orderq(odd) and the multiplicative order ofq modulo 2 ⁿ (n>1) be ?(2 ⁿ )/2. Ifn>3, thenq is odd number(prime or prime power) of the form 8m±3. Ifq=8m?3, then the ring $$R_{2^n } = F\left[ x \right]/< x^{2^n } - 1 > $$ has 2n primitive idempotents. The explicit expressions for these primitive idempotents are obtained and the minimal QR cyclic codes of length 2 ⁿ generated by these idempotents are completely described. Ifq=8m+3 then the expressions for the 2n?1 primitive idempotents ofR ₂ ⁿ are obtained. The generating polynomials and the upper bounds of the minimum distance of minimal QR cyclic codes generated by these 2n?1 idempotents are also obtained. The casen=2, 3 is dealt separately.     

Приближение всплесками и поперечники Фурье классов периодических функций многих переменных. II

Estimates sharp in order for Fourier widths of the classes $ B_{pq}^{sm} (\mathbb{T}^k ) $ and $ L_{pq}^{sm} (\mathbb{T}^k ) $ of Nikol??skii-Besov and Lizorkin-Triebel types, respectively, in the space $ L_r (\mathbb{T}^k ) $ are established for a certain range of the parameters s, p, q, r (here s ?? (0,??) ⁿ , 1 ??p, r, q ???, 1 ?? n ?? k, m = (m ₁, ??,m _n ) ?? ? ⁿ : m ₁ + ?? + m _n = k).