Spectral-Null Codes and Null Spaces of Hadamard Submatrices

Codes of length 2 ^m over {1, -1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of all have an rth order spectral null at zero frequency. Establishing the connection between and the parity-check matrix of Reed-Muller codes, the minimum distance of is obtained along with upper bounds on the redundancy of . An efficient algorithm is presented for encoding unconstrained binary sequences into .     

利用有限域上酉几何构作带仲裁的认证码

利用有限域上酉几何构作一类新的带仲裁的认证码,并且计算了所构作认证码的参数以及各种攻击成功的概率.     

循环码译码的Dixon结式方法

针对纠错码译码就是非线性方程组的求解问题,提出利用Dixon结式方法对译码方程进行消元以得到接收数据中的错位多项式.首先,根据纠错码的纠错能力和接收数据得到伴随式矩阵并通过该矩阵的秩确定接收码字中错误位的个数.然后,根据错位个数和伴随多项式构造译码方程.译码时,将其中一个错位变元作为隐藏变元,利用Dixon结式方法进行消元.最后,得到的Dixon结式就是关于隐藏变元的多项式.该多项式去掉多余因子后就是错位多项式,利用Chien搜索法即可求解出错误位置.当错位较多时,采用逐次计算结式的方法以筛除计算过程中的多余因子和重因子.另外,根据不同错位个数得到的错位多项式,提出了构造一类循环码错位多项式符号解的猜想,该猜想可以大大提高译码效率.实验验证了结式理论在纠错码译码方面的应用是有效的且有助于降低对芯片性能的要求.     

具有优良渐近参数的代数几何码

本文讨论了一类具有好的渐近参数的代数几何码.通过对除子类数、高次有理除子数以及代数几何码的参数分析,得到一类码其渐近界优于Gilbert-Varshamov界和Xing界.在这两个界的交点处,渐近界有所改进.     

Construction of optimal ternary constant weight codes via Bhaskar Rao designs

In this note, we consider a construction for optimal ternary constant weight codes (CWCs) via Bhaskar Rao designs (BRDs). The known existence results for BRDs are employed to generate many new optimal nonlinear ternary CWCs with constant weight 4 and minimum Hamming distance 5.     

Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP=λP for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1?r<k it is shown that Λk(A)⊆Λk−r(A+F) for any operator F with rank(F)?r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a (k−r)-dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λk(A) can be obtained as the intersection of Λk−r(A+F) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λk(A) are completely determined. Analogous results are obtained for Λ_∞(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that Λ_∞(A) is the intersection of all Λk(A) for k=1,2,…. If A−μI is not compact for all μ∈C, then the closure and the interior of Λ_∞(A) coincide with those of the essential numerical range of A. The situation for the special case when A−μI is compact for some μ∈C is also studied.     

关于循环阵列码

本文用代数观点来研究循环阵列码,证明了一般的阵列码是一些极小循环阵列码的直和,并且对极小循环阵列码给出了明确的刻画．当有限域的特征不整除群的阶时,给出了直接写出相应的多项式环的本原幂等元的方法,从而可以直接写出所有的极小循环码．     

Cyclic Codes Over\mathbb{Z}_{4} of Even Length

We determine the structure of cyclic codes over for arbitrary even length giving the generator polynomial for these codes. We determine the number of cyclic codes for a given length. We describe the duals of the cyclic codes, describe the form of cyclic codes that are self-dual and give the number of these codes. We end by examining specific cases of cyclic codes, giving all cyclic self-dual codes of length less than or equal to 14. San Ling - The research of the second named author is partially supported by research Grants MOE-ARF R-146-000-029-112 and DSTA R-394-000-011-422.     

Super-simple (ν, 5, 5) Designs

Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. In this article, we investigate the existence of a super-simple (ν, 5, 5) balanced incomplete block design and show that such a design exists if and only if ν ≡ 1 (mod 4) and ν ≥ 17 except possibly when ν = 21. Applications of the results to optical orthogonal codes are also mentioned. Research supported by NSERC grant 239135-01.