Parameterized Complexity of Satisfying Almost All Linear Equations over \mathbb{F}_{2}

The problem MaxLin2 can be stated as follows. We are given a system S of m equations in variables x ₁,…,x _n , where each equation $\sum_{i \in I_{j}}x_{i} = b_{j}$ is assigned a positive integral weight w _j and $b_{j} \in\mathbb{F}_{2}$ , I _j ?{1,2,…,n} for j=1,…,m. We are required to find an assignment of values in $\mathbb{F}_{2}$ to the variables in order to maximize the total weight of the satisfied equations. Let W be the total weight of all equations in S. We consider the following parameterized version of MaxLin2: decide whether there is an assignment satisfying equations of total weight at least W?k, where k is a nonnegative parameter. We prove that this parameterized problem is W[1]-hard even if each equation of S has exactly three variables and every variable appears in exactly three equations and, moreover, each weight w _j equals 1 and no two equations have the same left-hand side. We show the tightness of this result by proving that if each equation has at most two variables then the parameterized problem is fixed-parameter tractable. We also prove that if no variable appears in more than two equations then we can maximize the total weight of satisfied equations in polynomial time.     

u-Constacyclic codes over $${\mathbb {F}}_p+ u{\mathbb {F}}_p$$Fp+ uFp and their applications of constr ucting new non-binary q uant um codes

Structural properties of u-constacyclic codes over the ring \({\mathbb {F}}_p+u{\mathbb {F}}_p\) are given, where p is an odd prime and \(u^2=1\). Under a special Gray map from \({\mathbb {F}}_p+u{\mathbb {F}}_p\) to \({\mathbb {F}}_p^2\), some new non-binary quantum codes are obtained by this class of constacyclic codes.     

Quaternary Codes and Biphase Sequences from $$\mathbb{Z}_8 $$ -Codes

Composing the Carlet map with the inverse Gray map, a new family of cyclic quaternary codes is constructed from 5-cyclic -codes. This new family of codes is inspired by the quaternary Shanbag–Kumar–Helleseth family, a recent improvement on the Delsarte–Goethals family. We conjecture that these -codes are not linear. As applications, we construct families of low-correlation quadriphase and biphase sequences.     

Construction of punctured and extended quantum codes over GF(2)

On the basis of elementary transformation,we propose a new method for constructing a class of pure quantum codes [[n-i,2k-n+i,d-i]]2 and [[n+1,2k-n-1,d+1]]2 from a class of classical linear codes [n,k,d]2 that contain their dual codes.The construction process was based on the elementary algebra;the error-correcting performance of the quantum codes was analyzed according to the relationship between the parity-check matrix and the minimum distance of the classical linear codes;the encoding and decoding networks were constructed based on the stabilizer.The proposed method is simple,straightforward and easy to implement using a computer and other hardware system.The theoretical results showed that the method is practical for the construction of a class of quantum codes.     

New q-ary quantum MDS codes with distances bigger than $$\frac{ q}{2}$$

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474–1484, 2015) for known constructions. However, there have been constructed only a few q-ary quantum MDS \([[n,n-2d+2,d]]_q\) codes with minimum distances \(d>\frac{q}{2}\) for sparse lengths \(n>q+1\). In the case \(n=\frac{q^2-1}{m}\) where \(m|q+1\) or \(m|q-1\) there are complete results. In the case \(n=\frac{q^2-1}{m}\) while \(m|q^2-1\) is neither a factor of \(q-1\) nor \(q+1\), no q-ary quantum MDS code with \(d> \frac{q}{2}\) has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over \(\mathbf{F}_{q^2}\). Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form \(\frac{w(q^2-1)}{u}\) and minimum distances \(d > \frac{q}{2}\) are presented.     

The distance between the load and the body with three bi-manual lifting techniques

Studies were made of two different techniques of bi-manual lifting with bent legs (A and B) and a technique of lifting with the back bent and the Knees almost extended (C). With technique A, the trunk was almost vertical, while with B it was erect and more forward inclined and the heel of the front foot kept in contact with the support. Two healthy subject samples (n = 18 and n = 16 respectively) were studied, both employing a force platform; with the second sample the back muscles were also evaluated by electromyography. The distance Delta L between the lines of gravity of the body and the load at the start of the lifts was shortest with technique A and longest with C. This was true whether the position of the feet was chosen spontaneously or was identical for all three techniques. The distance of the load from the body during the lifting movement was directly related to the distance at the start of the lift: the further away the load was at initiation of the lift, the further away it remained throughout the rest of the lift. A request to lift as close to the load as possible had a positive effect in shortening Delta L , but the amount of previously received instruction in lifting technique did not correlate with the spontaneously chosen Delta L.     

有机锡配合物{[n-Bu2Sn(O2 CC5 H3 NCl)]2 O}2的合成、结构及量子化学研究

二正丁基氧化锡和2-氯-3-吡啶甲酸反应,合成2-氯-3-吡啶甲酸二正丁基锡配合物{[n-Bu2Sn(O2CC5H3NCl]2O}2.经X-射线衍射法测定了晶体结构.晶体属三斜晶系,空间群P-1,晶体学参数a=1.17841(9)nm,b=1.20811(9)nm,c=2.7460(2)nm,α=80.5330(10)°,β=84.1140(10)°,γ=64.2450(10)°,Z=2,V=3.4709(5)nm3,Dc=1.521 mg·m-3,μ(MoKa)=1 628 mm-1,F(000)=1592,R1=0.0430,wR2=0.1005.化合物是以Sn2O2构成的平面四元环为中心环的二聚体结构,锡原子均为五配位的畸变三角双锥形.用量子化学从头计算其结构,探讨配合物的稳定性、分子轨道能量以及一些前沿分子轨道的组成特征.     

Some high-rate linear codes over <Emphasis Type="Italic">GF</Emphasis>(5) and <Emphasis Type="Italic">GF</Emphasis>(7)

Let an [n, k, d] _q code be a linear code of length n, dimension k, and with minimum Hamming distance d over GF(q). The ratio R = k/n is called the rate of a code. In this paper, [62, 53, 6]₅, [62, 48, 8]₅, [71, 56, 8]₅, [124, 113, 6]₅, [43, 36, 6]₇, [33, 23, 7]₇, and [27, 18, 7]₇ high-rate codes and new codes with parameters [42, 14, 19]₅, [42, 15, 18]₅, [48, 13, 24]₅, [52, 12, 28]₅, [71, 15, 38]₅, [71, 16, 36]₅, [72, 12, 41]₅, [78, 10, 50]₅, [88, 11, 54]₅, [88, 13, 51]₅, [124, 14, 77]₅, [32, 12, 15]₇, [32, 10, 17]₇, [36, 10, 20]₇, and [48, 10, 29]₇ are constructed. The codes with parameters [62, 53, 6]₅ and [43, 36, 6]₇ are optimal.     

Branching Time Logics \mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha } with Operations Until and Since Based on Bundles of Integer Numbers, Logical Consecutions, Deciding Algorithms

This paper is intended as an attempt to describe logical consequence in branching time logics. We study temporal branching time logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ which use the standard operations Until and Next and dual operations Since and Previous (LTL, as standard, uses only Until and Next). Temporal logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ are generated by semantics based on Kripke/Hinttikka structures with linear frames of integer numbers $\mathcal {Z}$ with a single node (glued zeros). For $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ , the permissible branching of the node is limited by α (where 1≤α≤ω). We prove that any logic $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ is decidable w.r.t. admissible consecutions (inference rules), i.e. we find an algorithm recognizing consecutions admissible in $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ . As a consequence, it implies that $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ itself is decidable and solves the satisfiability problem.