The Gray images of $$$$ constacyclic codes over $$F_{2^m}[u]/\langle u^{k} \rangle $$F2m[u]/⟨uk⟩

Let \(R_{k}\) denote the polynomial residue ring \(F_{2^m}[u]/\langle u^{k} \rangle \), where \(2^{j-1}+1\le k\le 2^{j}\) for some positive integer \(j\). Motivated by the work in [1], we introduce a new Gray map from \(R_{k}\) to \(F_{2^m}^{2^{j}}\). It is proved that the Gray image of a linear \((1+u)\) constacyclic code of an arbitrary length \(N\) over \(R_{k}\) is a distance invariant linear cyclic code of length \(2^{j}N\) over \(F_{2^m}\). Moreover, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over \(F_{2}\) and \(F_{4}\) are constructed under this Gray map.     

环IF_pm[u]/〈u~a〉上一类常循环码

本文研究有限链环上一类λ-常循环码.利用x~n-1在R_a[x]上可唯一分解为两两互素的首一基本不可约多项式乘积,刻画了R_a中长为p~sn的所有λ-常循环码,推广了开晓山等人在文献[4]中的结果.     

Characterizations of some two-dimensional cyclic codes correspond to the ideals of F[x,y]/

Two-dimensional cyclic code is one of the natural generalizations of cyclic code. In this paper we study the algebraic structure of some two-dimensional cyclic codes and their dual codes.     

On the Greatest Common Di visor of u − 1 and v − 1 with u and v Near ${\cal S}$- units

In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (aⁿ − 1,bⁿ − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f₁(x),g(x),g₁(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality holds for all but finitely many positive integers n.     

On the distinctness of modular reductions of primitive sequences over Z/(232−1)

This paper studies the distinctness of modular reductions of primitive sequences over ${\mathbf{Z}/(2^{32}-1)}$ . Let f(x) be a primitive polynomial of degree n over ${\mathbf{Z}/(2^{32}-1)}$ and H a positive integer with a prime factor coprime with 2³²?1. Under the assumption that every element in ${\mathbf{Z}/(2^{32}-1)}$ occurs in a primitive sequence of order n over ${\mathbf{Z}/(2^{32}-1)}$ , it is proved that for two primitive sequences ${\underline{a}=(a(t))_{t\geq 0}}$ and ${\underline{b}=(b(t))_{t\geq 0}}$ generated by f(x) over ${\mathbf{Z}/(2^{32}-1), \underline{a}=\underline{b}}$ if and only if ${a\left( t\right) \equiv b\left( t\right) \bmod{H}}$ for all t ≥ 0. Furthermore, the assumption is known to be valid for n between 7 and 100, 000, the range of which is sufficient for practical applications.     

On Diophantine equation 3a 2 x 4 − By 2 = 1

Bumby proved that the only positive solutions to the quartic Diophantine equation 3x ⁴ ? 2y ² = 1 are (x, y) = (1, 1), (3, 11). In this paper, we extend this result and prove that if the class number of the field ${{\rm Q}(\sqrt{1-3a^{2}})}Bumby proved that the only positive solutions to the quartic Diophantine equation 3x ⁴ − 2y ² = 1 are (x, y) = (1, 1), (3, 11). In this paper, we extend this result and prove that if the class number of the field Q(?{1-3a²}){{\rm Q}(\sqrt{1-3a^{2}})} is not divisible by 2, the equation 3a ² x ⁴ − By ² = 1 has at most two solutions. However, both solutions occur in only one case, a = 1, b = 2, as solved by Bumby. The proof utilizes the law of quadratic reciprocity that seems very rare in solving Diophantine equations, and the solution will be also obtained effectively through the proof when it exists.     

On the Multiplicative Orders of γ and γ+γ−1 over finite fields

Using bounds of character sums we show that one of the open questions about the possible relation between the multiplicative orders of γ and γ+γ⁻¹ has a negative answer. In fact we show that in some sense the multiplicative orders of these elements are independent.     

On the diophantine equation X 2 − (1 + a 2)Y 4 = −2a

Let a≥1 be an integer.In this paper,we will prove the equation in the title has at most three positive integer solutions.     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

Group Laws [x,y −1] ≡ u(x,y) and Varietal Properties

Let F = ?x, y? be a free group. It is known that the commutator [x, y ^?1] cannot be expressed in terms of basic commutators, in particular in terms of Engel commutators. We show that the laws imposing such an expression define specific varietal properties. For a property 𝒫 we consider a subset U(𝒫) ? F such that every law of the form [x, y ^?1] ≡ u, u ∈ U(𝒫) provides the varietal property 𝒫. For example, we show that each subnormal subgroup is normal in every group of a variety 𝔙 if and only if 𝔙 satisfies a law of the form [x, y ^?1] ≡ u, where u ∈ [F′, ?x?].     

On the construction of [q 4 + q 2 − q, 5,q 4 − q 3 + q 2 − 2q; q]-codes meeting the Griesmer bound

It is unknown whether or not there exists an [87, 5, 57; 3]-code. Such a code would meet the Griesmer bound. The purpose of this paper is to give a constructive proof of the existence of [q ⁴ + q ² – q, 5, q ⁴ – q ³ + q ² – 2q; q]-codes for any prime power q 3. As a special case, it is shown that there exists an [87, 5, 57; 3]-code with weight enumerator 1 + 156z ³⁷ + 82z ⁶⁰ + 2z ⁶³ + 2z ⁷⁸. The new construction settles an open problem due to Hill and Newton [10].     

Sections of U(2n)/Sp(s) over S4n−1

We describe the sections of U(2n)/Sp(s) over S⁴ⁿ⁻¹ in terms of the sections of the symplectic Stiefel manifold Sp(n)/Sp(s) and we express the orders of obstructions to sectioning U(2n)/Sp(s) in terms of orders of obstructions to sectioning Sp(n)/Sp(s). In certain cases we give the exact values of these orders.     

Projective line over the finite quotient ring GF(2)[x]/〈x3 ? x〉 and quantum entanglement: Theoretical background

We consider the projective line over the finite quotient ring R_⋄ ≡ GF(2)[x]/〈x ³ − x〉. The line is endowed with 18 points, spanning the neighborhoods of three pairwise distant points. Because R_⋄ is not a local ring, the neighbor (or parallel) relation is not an equivalence relation, and the sets of neighbors for two distant points hence overlap. There are nine neighbors of any point on the line, forming three disjoint families under the reduction modulo either of the two maximal ideals of the ring. Two of the families contain four points each, and they swap their roles when switching from one ideal to the other, the points in one family merging with (the image of) the point in question and the points in the other family passing in pairs into the remaining two points of the associated ordinary projective line of order two. The single point in the remaining family passes to the reference point under both maps, and its existence stems from a nontrivial character of the Jacobson radical of the ring. The quotient ring is isomorphic to GF(2) ⊗ GF(2). The projective line over features nine points, each of them surrounded by four neighbors and four distant points, and any two distant points share two neighbors. We surmise that these remarkable ring geometries are relevant for modeling entangled qubit states, which we will discuss in detail in Part II of this paper. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 44–53, April, 2007.