Beyond stabilizer codes .I. Nice error bases

Nice error bases have been introduced by Knill (1996) as a generalization of the Pauli basis. These bases are shown to be projective representations of finite groups. We classify all nice error bases of small degree, and all nice error bases with Abelian index groups. We show that, in general, an index group of a nice error basis is necessarily solvable.     

量子稳定子码的概率译码

为了提高量子稳定子码的译码速率,提出了一种基于校验矩阵的量子概率译码算法。通过选择具有最小量子权重的算子作为差错算子来减少译码出错概率,通过预先构造量子标准阵列来缩短译码时间。与已有的量子最大似然译码算法相比,该算法对简并码和非简并码采用统一的译码方式,从而提高了简并码的译码可靠性。此外,算法不需要预先寻找差错算子对应的向量空间的基,因此算法复杂度更小。     

The construction method of stabilizer codes for continuous variables

The paper analyzes the basic principles of stabilizer codes, focusing on how to construct stabilizer codes for achieving the continuous-variable quantum error correction. Stabilizer codes can be used in the reconciliation of continuous-variable quantum key distribution system. The construction method of stabilizer codes is very important and it can be turned into finding the check matrix for stabilizer codes. In this paper, a new algorithm called region elimination algorithm for finding the check matrix of stabilizer codes was presented which can seek the voluntary check matrix for continuous-variable stabilizer codes within 8 bit code length quickly and effectively, and it was simulated by Visual C++. The algorithm is mainly realized by initializing search region, reducing the search region and then keeping searching till finding all the commuting generators. The finding of check matrix of stabilizer codes lays important foundations for the further development of stabilizer codes in the continuous-variable quantum key distribution.     

Asymptotically dense spherical codes .II. laminated spherical codes

For pt. I see ibid., vol.43, no.6, p.1774-85, 1997. New spherical codes called laminated spherical codes are constructed in dimensions 2-49 using a technique similar to the construction of laminated lattices. Each spherical code is recursively constructed from existing spherical codes in one lower dimension. Laminated spherical codes outperform the best known spherical codes in the minimum distance sense for many code sizes. The density of a laminated spherical code approaches the density of the laminated lattice in one lower dimension, as the minimum distance approaches zero. In particular, the three-dimensional laminated spherical code is asymptotically optimal, in the sense that its density approaches the Fejes Toth (1959) upper bound as the minimum distance approaches zero. Laminated spherical codes perform asymptotically as well as wrapped spherical codes in those dimensions where laminated lattices are optimal sphere packings     

Rotational invariance of trellis codes. II. Group codes anddecoders

For pt.I see ibid., vol.42, no.3, p.751-65 (1996). In Part I, general results on rotationally invariant codes and encoders were derived assuming no algebraic structure. In Part II, trellis codes based on group systems are considered as a special case for which code and encoder constructions are particularly simple. Rotational invariance is expressed as an algebraic constraint on a group code, and algebraic constructions are found for both “absorbed precoder” encoders and for encoders with separate differential precoders. Finally, the various encoder forms used to achieve rotational invariance are compared based on their performance on an AWGN channel     

nd-convolutional codes. II. Structural analysis

For pt.I see ibid., vol.43, no.2, p.558-75 (1997). The structural properties of a noncoherent coded system, which incorporates convolutional codes in conjunction with multiple symbol noncoherent detection, is presented in this second part of a two-part paper, where the performance analysis was provided in Part I. These convolutional codes are referred to as nd-convolutional codes and they provide a general framework for various noncoherent coding systems, including differential systems, for several practical models of the carrier phase. The exponential rate in which the error probability decays to zero, derived in Part I of the paper, is used here to obtain the free equivalent distance of nd-codes, which is the single parameter dominating the error performance at large signal-to-noise ratios. The free equivalent distance is upper-bounded by the free nd-distance, which constitutes a more convenient and practical parameter to work with, and it is the basis for a computer search for optimal nd-codes. The resultant codes of the computer search are compared to codes which are optimal for coherent detection, and it is verified that the latter codes are not necessarily optimal for noncoherent detection since they exhibit in many cases a relatively small nd-distance. The ambiguity problem, inherent to noncoherent systems, is also treated in this paper in the general framework of nd-catastrophic codes, and necessary and sufficient conditions for catastrophic error propagation are identified     

Exponential sums and Goppa codes. II

For pt.I, see Proc. AMS, vol.III, p.523-31 (1991). The minimum distance of a Goppa code is found when the length of code satisfies a certain inequality on the degree of the Goppa polynomial. In order to do this, conditions are improved on a theorem of E. Bombieri (1966). This improvement is used also to generalize a previous result on the minimum distance of the dual of a Goppa code. This approach is generalized and results are obtained about the parameters of a class of subfield subcodes of geometric Goppa codes; in other words, the covering radii are estimated, and further, the number of information symbols whenever the minimum distance is small in relation to the length of the code is found. Finally, a bound on the minimum distance of the dual code is discussed     

A note on type II convolutional codes

The result of a search for the world's second type II (doubly-even and self-dual) convolutional code is reported. A rate R=4/8, 16-state, time-invariant, convolutional code with free distance d_free=8 was found to be type II. The initial part of its weight spectrum is better than that of the Golay convolutional code (GCC). Generator matrices and path weight enumerators for some other type II convolutional codes are given. By the “wrap-around” technique tail-biting versions of (32, 18, 8) Type II block codes are constructed     

Life Beyond Bases: The Advent of Frames (Part II)

While the first part of this article presented most of the basic theoretical developments of frames, this part is more user friendly. It covers a large number of known frame families (harmonic tight frames, equiangular frames, unit-norm tight frame, Gabor frames, cosine-modulated frames, double-density frames, multidimensional frames, filter bank frame) as well as those applications where frames made a difference.     

Error-trellises for convolutional codes .II. Decoding methods

For pt. I see ibid. vol.46, p.1592-1601 (1998). Soft-decision maximum-likelihood decoding of convolutional codes over GF(q) can be accomplished via searching through an error-trellis for the least weighing error sequence. The error-trellis is obtained by a syndrome-based construction. Its structure lends itself particularly well to the application of expedited search procedures. The method to carry out such error-trellis-based decoding is formulated by four algorithms. Three of these algorithms are aimed at reducing the worst case computational complexity, whereas by applying the fourth algorithm, the average computational complexity is reduced under low to moderate channel wise level. The syndrome decoder achieves substantial worst case and average computational gains in comparison with the conventional maximum-likelihood decoder, namely the Viterbi decoder, which searches for the most likely codeword directly within the code     

Lexicographic codes: Error-correcting codes from game theory

Lexicographic codes, or lexicodes, are defined by various versions of the greedy algorithm. The theory of these codes is closely related to the theory of certain impartial games, which leads to a number of surprising properties. For example, lexicodes over an alphabet of sizeB=2^{a}are closed under addition, while ifB = 2^{2^{a}}the lexicodes are closed under multiplication by scalars, where addition and multiplication are in the nim sense explained in the text. Hamming codes and the binary Golay codes are lexieodes. Remarkably simple constructions are given for the Steiner systemsS(5,6,12)andS(5,8,24). Several record-breaking constant weight codes are also constructed.     

P2 codes: pragmatic trellis codes utilizing puncturedconvolutional codes

A single convolutional code of fixed rate can be punctured to form a class of higher rate convolutional codes. The authors extend this pragmatic approach to the case where the core of the trellis decoder is a Viterbi decoder for a punctured version of the de facto standard, rate 1/2 convolutional code     

On Type II codes over F4

Previously, Type II codes over F₄ have been introduced as Euclidean self-dual codes with the property that all Lee weights are divisible by four. In this paper, a number of properties of Type II codes are presented. We construct several extremal Type II codes and a number of extremal Type I codes. It is also shown that there are seven Type II codes of length 12, up to permutation equivalence     

Near-optimum decoding of product codes: block turbo codes

This paper describes an iterative decoding algorithm for any product code built using linear block codes. It is based on soft-input/soft-output decoders for decoding the component codes so that near-optimum performance is obtained at each iteration. This soft-input/soft-output decoder is a Chase decoder which delivers soft outputs instead of binary decisions. The soft output of the decoder is an estimation of the log-likelihood ratio (LLR) of the binary decisions given by the Chase decoder. The theoretical justifications of this algorithm are developed and the method used for computing the soft output is fully described. The iterative decoding of product codes is also known as the block turbo code (BTC) because the concept is quite similar to turbo codes based on iterative decoding of concatenated recursive convolutional codes. The performance of different Bose-Chaudhuri-Hocquenghem (BCH)-BTCs are given for the Gaussian and the Rayleigh channel. Performance on the Gaussian channel indicates that data transmission at 0.8 dB of Shannon's limit or more than 98% (R/C>0.98) of channel capacity can be achieved with high-code-rate BTC using only four iterations. For the Rayleigh channel, the slope of the bit-error rate (BER) curve is as steep as for the Gaussian channel without using channel state information     

Linear inequalities for covering codes. II. Triple coveringinequalities

For Pt.I, see ibid., vol.37, no.3, p.573-82 (May 1991). The linear inequality method for covering codes is generalized. This method reduces the study of covering codes to the study of some local covering problems. One of these problems, the 1-3 covering system, is formulated and studied in detail. The results for this local covering problem lead to new linear inequalities satisfied by covering codes, which are used to obtain numerous new lower bounds on K(n, R) and t[n, k]     

A stabilizer for cavity-dumped Nd:YAG

Relaxation oscillations induced by cavity dumping in a Nd:YAG laser have been suppressed by an intracavity frequency doubler, thereby increasing the useful output of this laser for communications purposes by an order of magnitude. The additional loss induced by frequency doubling was negligible even at high damping constants.     

Generalized H-codes and type Ⅱcodes over GF（4）

The typeⅡ codes have been studied widely in applications since their appearance. With analysis of the algebraic structure of finite field of order 4 （i.e., GF（4））, some necessary and sufficient conditions that a generalized H-code （i.e., GH-code） is a type Ⅱ code over GF（4） are given in this article, and an efficient and simple method to generate type Ⅱ codes from GH-codes over GF（4） is shown. The conclusions further extend the coding theory of type Ⅱ.     

Augmented product codes and lattices: Reed-Muller codes and Barnes-Wall lattices

This paper concerns the construction of the so-called augmented product codes and augmented product lattices. These are obtained by augmenting product codes or product lattices from certain classes thus obtaining higher dimensional codes or lattices from the same class, respectively. Certain properties of the augmented product construction are derived, and specific construction examples are given. In particular, it is shown that the Reed-Muller codes, the Golay code, the Barnes-Wall lattices, as well as the Leech lattice all have various augmented product constructions.