Quantum shadow enumerators

In a previous paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-02, 1997) define two “weight enumerators” for quantum error-correcting codes, connected by a MacWilliams (1977) transform, and use them to give a linear-programming bound for quantum codes. We extend their work by introducing another enumerator, based on the classical theory of shadow codes, that tightens their bounds significantly. In particular, nearly all of the codes known to be optimal among additive quantum codes (codes derived from orthogonal geometry) can be shown to be optimal among all quantum codes. We also use the shadow machinery to extend a bound on additive codes to general codes, obtaining as a consequence that any code of length, can correct at most [(n+1)/6] errors     

Polynomial invariants of quantum codes

The weight enumerators (Shor and Laflamme 1997) of a quantum code are quite powerful tools for exploring its structure. As the weight enumerators are quadratic invariants of the code, this suggests the consideration of higher degree polynomial invariants. We show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one-to-one correspondence with S_kⁿ. We then present a number of equations and inequalities in these invariants; in particular, we give a higher order generalization of the shadow enumerator of a code, and prove that its coefficients are nonnegative. We also prove that the quartic invariants of a ((4, 4, 2))₂ code are uniquely determined, an important step in a proof that any ((4, 4, 2))₂ code is additive (Rains 1999)     

On Z4-duality

Recently the notion on binary codes called Z₄-linearity was introduced. This notion explains why Kerdock codes and Delsarte-Goethals codes admit formal duals in spite of their nonlinearity. The “Z₄-duals” of these codes (called “Preparata” and “Goethals” codes) are new nonlinear codes which admit simpler decoding algorithms than the previously known formal duals (the generalized Preparata and Goethals codes). We prove, by using the notion of exact weight enumerator, that the relationship between any Z₄-linear code and its Z₄ -dual is stronger than the standard formal duality and we deduce the weight enumerators of related generalized codes     

Quantum error detection .I. Statement of the problem

This paper is devoted to the problem of error detection with quantum codes. We show that it is possible to give a consistent definition of the undetected error event. To prove this, we examine possible problem settings for quantum error detection. Our goal is to derive a functional that describes the probability of undetected error under natural physical assumptions concerning transmission with error detection with quantum codes. We discuss possible transmission protocols with stabilizer and unrestricted quantum codes. The set of results proved in the paper shows that in all the cases considered the average probability of undetected error for a given code is essentially given by one and the same function of its weight enumerators. We examine polynomial invariants of quantum codes and show that coefficients of Rains's (see ibid., vol44, p.1388-94, 1998) “unitary weight enumerators” are known for classical codes under the name of binomial moments of the distance distribution. As in the classical situation, these enumerators provide an alternative expression for the probability of undetected error     

Quantum error detection .II. Bounds

For pt.I see ibid., vol.46, no.3, p.778-88 (2000). In Part I of this paper we formulated the problem of error detection with quantum codes on the depolarizing channel and gave an expression for the probability of undetected error via the weight enumerators of the code. In this part we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent. The lower (existence) bound is proved for stabilizer codes by a counting argument for classical self-orthogonal quaternary codes. Upper bounds are proved by linear programming. First we formulate two linear programming problems that are convenient for the analysis of specific short codes. Next we give a relaxed formulation of the problem in terms of optimization on the cone of polynomials in the Krawtchouk basis. We present two general solutions of the problem. Together they give an upper bound on the exponent of undetected error. The upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1     

Translates of linear codes over Z4

We give a method to compute the complete weight distribution of translates of linear codes over Z₄. The method follows known ideas that have already been used successfully by others for Hamming weight distributions. For the particular case of quaternary Preparata codes, we obtain that the number of distinct complete weights for the dual Preparata codes and the number of distinct complete coset weight enumerators for the Preparata codes are both equal to ten, independent of the code length     

On factor graphs and the Fourier transform

We introduce the concept of convolutional factor graphs, which represent convolutional factorizations of multivariate functions, just as conventional (multiplicative) factor graphs represent multiplicative factorizations. Convolutional and multiplicative factor graphs arise as natural Fourier transform duals. In coding theory applications, algebraic duality of group codes is essentially an instance of Fourier transform duality. Convolutional factor graphs arise when a code is represented as a sum of subcodes, just as conventional multiplicative factor graphs arise when a code is represented as an intersection of supercodes. With auxiliary variables, convolutional factor graphs give rise to "syndrome realizations" of codes, just as multiplicative factor graphs with auxiliary variables give rise to "state realizations." We introduce normal and co-normal extensions of a multivariate function, which essentially allow a given function to be represented with either a multiplicative or a convolutional factorization, as is convenient. We use these function extensions to derive a number of duality relationships among the corresponding factor graphs, and use these relationships to obtain the duality properties of Forney graphs as a special case.     

A family of low-complexity binary linear codes for Bluetooth and BLAST signaling applications

We construct a family of linear binary block codes that are useful for Bluetooth, OFDM and BLAST applications. These codes are derived from ordinary block repetition codes using cyclic shifts of the input information vector which greatly simplifies encoding. We find several sets of cyclic shifts that constrain the search for good codes. We consider code lengthening and the input-output weight enumerators. We show that the codes are good candidates for low-power, low-cost and high data-rate applications using fixed code rate and variable codeword length, or adaptive coding with variable minimum Hamming distance. We propose a parallel structure of the encoder well-suited to OFDM and BLAST systems. Finally, we give an example of code design for use in retransmission schemes, and another example of a concatenated rate 2/3 code well-suited to the Bluetooth system.     

On the input-output weight enumerators of product accumulate codes

In this letter, a generalized MacWilliams transform that relates the input-redundancy weight enumerator of a systematic binary linear block code to that of its dual code is first presented. Based on this transform, the input-output weight enumerators of direct-product single-parity-check codes and the type-II product-accumulate codes are then derived, and used to analyze the asymptotic bit error performance of these codes.     

Jacobi polynomials for singly even self-dual codes and the coveringradius problems

In this correspondence, we develop a method to determine the complete coset weight distributions of the class of singly even self-dual binary codes. Our basic tool is the Jacobi polynomials for the code. It describes and controls the coset weight enumerators. As the results of our present method, we give the complete coset weight distributions of some extremal singly even self-dual codes of lengths 14, 22, 32, 36, and 40, respectively. We give the generator matrices of the used codes of lengths 36 and 40, respectively     

环Fp+vFp+v2Fp上线性码的各种重量计数器及其MacWilliams恒等式

首先给出了环R=F_p+vF_p+v²F_p上线性码及其对偶码的结构及其Gray象的性质.定义了环R上线性码的各种重量计数器并讨论了它们之间的关系,特别的,确定了该环上线性码与其对偶码之间关于完全重量计数器的MacWilliams恒等式,利用该恒等式,进一步建立了该环上线性码与其对偶码之间的一种对称形式的MacWilliams恒等式.最后,利用该对称形式的MacWilliams恒等式得到了该环上的Hamming重量计数器和Lee重量计数器的MacWilliams恒等式,利用不同的方法推广了文献[7]中的结果.     

New extremal binary [44,22,8] codes

One of the problems in coding theory is constructing self-dual codes whose weight enumerators are not yet known to exist. We use the concept of neighbors and construct extremal binary self-dual [44,22,8] codes whose weight enumerators are not yet known to exist.     

The third support weight enumerators of the doubly-even, self-dual [32,16,8] codes

We present combinatorial methods for computing the third support weight enumerators of the five doubly-even, self-dual [32,16,8] codes. The methods exploit relationships that exist between support weight enumerators and complete coset weight enumerators of a self-dual code.     

On the modeling of DCT and subband image data for compression

Image subband and discrete cosine transform coefficients are modeled for efficient quantization and noiseless coding. Quantizers and codes are selected based on Laplacian, fixed generalized Gaussian, and adaptive generalized Gaussian models. The quantizers and codes based on the adaptive generalized Gaussian models are always superior in mean-squared error distortion performance but, generally, by no more than 0.08 bit/pixel, compared with the much simpler Laplacian model-based quantizers and noiseless codes. This provides strong motivation for the selection of pyramid codes for transform and subband image coding.     

Cubic self-dual binary codes

We study binary self-dual codes with a fixed point free automorphism of order three. All binary codes of that type can be obtained by a cubic construction that generalizes Turyn's. We regard such "cubic" codes of length 3/spl lscr/ as codes of length /spl lscr/ over the ring F/sub 2//spl times/F/sub 4/. Classical notions of Type II codes, shadow codes, and weight enumerators are adapted to that ring. Two infinite families of cubic codes are introduced. New extremal binary codes in lengths /spl les/ 66 are constructed by a randomized algorithm. Necessary conditions for the existence of a cubic [72,36,16] Type II code are derived.     

Complete joint weight enumerators and self-dual codes

We define the complete joint weight enumerator in genus g for codes over /spl Zopf//sub 2k/ and use it to study self-dual codes and their shadows. These weight enumerators are related to the theta series of the associated lattices and Siegel and Jacobi forms are formed from these series.     

环Z4+uZ4线性码关于李重量的一类MacWilliams恒等式

MacWilliams恒等式是研究线性码及其对偶码的码字重量分布的一个非常有用的工具,而码字的重量分布的研究是编码研究中一个非常重要的研究方向.本文定义了环Z4+uZ4上长度为n的线性码的m-层李重量计数器,给出了环Z4+uZ4上长度为n的线性码关于李重量的一类MacWilliams恒等式.证明了该等式是生成矩阵在环Z4+uZ4上的环GR(4,m)+uGR(4,m)上线性码关于李重量的MacWilliams恒等式.进一步,利用Krawtchouk多项式,获得了环Z4+uZ4上长度为n的线性码的等价形式MacWilliams恒等式.     

A Unified Method for Determining the Weight Enumerators of Binary Extended Quadratic Residue Codes

This letter proposes an improved and unified method to determine the weight enumerators of binary extended quadratic residue (EQR) codes. It is faster than the previous methods for some of binary EQR codes. Moreover, all the results for the weight enumerators of binary EQR codes are listed.     

一种无纠缠态的量子秘密共享协议

秘密共享是指将一个秘密按适当的方式进行隐藏或拆分,只有若干个参与者一同协作才能恢复该秘密,该技术在云计算领域中能够确保信息安全和数据保密.提出了一种不使用纠缠态的量子秘密共享协议,通过使用量子密码算法确保系统的安全性.相比其他的秘密共享协议,该协议具有以下优点:与传统的基于数论的秘密共享协议相比,本协议由于使用量子通信的技术,从而能够有效抵抗Shor算法攻击;相比其他的量子秘密共享协议,由于本协议没有使用量子纠缠态,在技术程度上更容易实现;如果存在攻击者或恶意的参与者,该协议能够在秘密恢复过程中迅速发现,避免恢复错误的秘密.     

The fast decoding of Reed-Solomon codes using Fermat theoretic transforms and continued fractions

It is shown that Reed-Solomon (RS) codes can be decoded by using a fast Fourier transform (FFT) algorithm over finite fieldsGF(F_{n}), whereF_{n}is a Fermat prime, and continued fractions. This new transform decoding method is simpler than the standard method for RS codes. The computing time of this new decoding algorithm in software can be faster than the standard decoding method for RS codes.