Circulant Codes Research Articles

Code Diversity in Multiple Antenna Wireless Communication

The standard approach to the design of individual space-time codes is based on optimizing diversity and coding gains. This geometric approach leads to remarkable examples, such as perfect space-time block codes, for which the complexity of Maximum Likelihood (ML) decoding is considerable. Code diversity is an alternative and complementary approach where a small number of feedback bits are used to select from a family of space-time codes. Different codes lead to different induced channels at the receiver, where Channel State Information (CSI) is used to instruct the transmitter how to choose the code. This method of feedback provides gains associated with beamforming while minimizing the number of feedback bits. It complements the standard approach to code design by taking advantage of different (possibly equivalent) realizations of a particular code design. Feedback can be combined with sub-optimal low complexity decoding of the component codes to match ML decoding performance of any individual code in the family. It can also be combined with ML decoding of the component codes to improve performance beyond ML decoding performance of any individual code. One method of implementing code diversity is the use of feedback to adapt the phase of a transmitted signal as shown for 4 by 4 Quasi-Orthogonal Space-Time Block Code (QOSTBC) and multi-user detection using the Alamouti code. Code diversity implemented by selecting from equivalent variants is used to improve ML decoding performance of the Golden code. This paper introduces a family of full rate circulant codes which can be linearly decoded by fourier decomposition of circulant matrices within the code diversity framework. A 3 by 3 circulant code is shown to outperform the Alamouti code at the same transmission rate.

Skew Hadamard designs and their codes

Skew Hadamard designs (4n – 1, 2n – 1, n – 1) are associated to order 4n skew Hadamard matrices in the natural way. We study the codes spanned by their incidence matrices A and by I + A and show that they are self-dual after extension (resp. extension and augmentation) over fields of characteristic dividing n. Quadratic Residues codes are obtained in the case of the Paley matrix. Results on the p-rank of skew Hadamard designs are rederived in that way. Codes from skew Hadamard designs are classified. An optimal self-dual code over GF(5) is rediscovered in length 20. Six new inequivalent [56, 28, 16] self-dual codes over GF(7) are obtained from skew Hadamard matrices of order 56, improving the only known quadratic double circulant code of length 56 over GF(7).

Asymptotic Improvement of the Gilbert–Varshamov Bound for Linear Codes

The Gilbert-Varshamov (GV) bound states that the maximum size A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (n, d) of a binary code of length n and minimum distance d satisfies A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (n, d)ges2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> /V(n, d-1) where V(n, d)=Sigma <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> ( <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) stands for the volume of a Hamming ball of radius d. Recently, Jiang and Vardy showed that for binary nonlinear codes this bound can be improved to A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (n, d)gescn2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> /(V(n, d-1)) for c a constant and d/nges0.499. In this paper, we show that certain asymptotic families of linear binary [n, n/2] random double circulant codes satisfy the same improved GV bound.

Fast neural network surrogates for very high dimensional physics-based models in computational oceanography

We present neural network surrogates that provide extremely fast and accurate emulation of a large-scale circulation model for the coupled Columbia River, its estuary and near ocean regions. The circulation model has O(10(7)) degrees of freedom, is highly nonlinear and is driven by ocean, atmospheric and river influences at its boundaries. The surrogates provide accurate emulation of the full circulation code and run over 1000 times faster. Such fast dynamic surrogates will enable significant advances in ensemble forecasts in oceanography and weather.

Double circulant codes from two class association schemes

Two class association schemes consist of either strongly regular graphs (SRG) or doubly regular tournaments (DRT). We construct self-dual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and Gaborit Quadratic Double Circulant codes (QDC). As new examples SRG's give 4 (resp. 5) new Type I (resp. Type II) [72, 36, 12] codes. We construct a [200, 100, 12] Type II code invariant under the Higman-Sims group, a [200, 100, 16] Type II code invariant under the Hall-Janko group, and more generally self-dual binary codes attached to rank three groups.

EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

Near-Extremal Formally Self-Dual Even Codes of Lengths 24 and 32

The weight enumerator of a formally self-dual even code is obtained by the Gleason theorem. Recently, Kim and Pless gave some restrictions on the possible weight enumerators of near-extremal formally self-dual even codes of length divisible by eight. In this paper, the weight enumerators for which there is a near-extremal formally self-dual even code are completely determined for lengths 24 and 32, by constructing new near-extremal formally self-dual codes. We also give a classification of near- extremal double circulant codes of lengths 24 and 32.

On the Weight Enumerators of Duadic and Quadratic Residue Codes

In this correspondence, we compute the weight enumerators of various quadratic residue codes over F/sub 2/ and F/sub 3/, together with certain codes of related families like the duadic and the quadratic double circulant codes. We use a parallel algorithm to find the number of codewords of a given (not too high) weight, from which we deduce by usual classical methods for self-dual and formally self-dual codes over F/sub 2/ and F/sub 3/ their associated, previously unknown, weight enumerators. We compute weight enumerators for lengths as high as 152 for binary codes and 96 for ternary codes.

Hardware system of the Earth Simulator

The Earth Simulator (ES), developed under the Japanese government’s initiative “Earth Simulator project”, is a highly parallel vector supercomputer system. In this paper, an overview of ES, its architectural features, hardware technology and the result of performance evaluation are described. In May 2002, the ES was acknowledged to be the most powerful computer in the world: 35.86 teraflop/s for the LINPACK HPC benchmark and 26.58 teraflop/s for an atmospheric general circulation code (AFES). Such a remarkable performance may be attributed to the following three architectural features; vector processor, shared-memory and high-bandwidth non-blocking interconnection crossbar network. The ES consists of 640 processor nodes (PN) and an interconnection network (IN), which are housed in 320 PN cabinets and 65 IN cabinets. The ES is installed in a specially designed building, 65 m long, 50 m wide and 17 m high. In order to accomplish this advanced system, many kinds of hardware technologies have been developed, such as a high-density and high-frequency LSI, a high-frequency signal transmission, a high-density packaging, and a high-efficiency cooling and power supply system with low noise so as to reduce whole volume of the ES and total power consumption. For highly parallel processing, a special synchronization means connecting all nodes, Global Barrier Counter (GBC), has been introduced.

Binary optimal linear rate 1/2 codes

In this paper, we complete the classification of binary optimal linear [ n, n/2] codes up to length 28. We have determined that there are 1535 inequivalent [14,7,4] codes, 1682 inequivalent [20,10,6] codes, and 3 inequivalent [26,13,7] codes. In addition, we show that the smallest length for which an optimal rate 1 2 code has a trivial automorphism group is 14. Double circulant codes which attain the highest known minimum distance are given for lengths up to 64.

Two-band wavelets and filterbanks over finite fields with connections to error control coding

Recently, we have developed a new framework to study error-control coding using finite-field wavelets and filterbanks (FBs). This framework reveals a rich set of signal processing techniques that can be exploited to investigate new error correcting codes and to simplify encoding and decoding techniques for some existing ones. The paper introduces the theory of wavelet decompositions of signals in vector spaces defined over Galois fields. To avoid the limitations of the number theoretic Fourier transform, our wavelet transform relies on a basis decomposition in the time rather than the frequency domain. First, by employing a symmetric, nondegenerate canonical bilinear form, we obtain a necessary and sufficient condition that the basis functions defined over finite fields must satisfy in order to construct an orthogonal wavelet transform. Then, we present a design methodology to generate the mother wavelet and scaling function over finite fields by relating the wavelet transform to two-channel paraunitary (PU) FBs. Finally, we describe the application of this transform to the construction of error correcting codes. In particular, we give examples of double circulant codes that are generated by wavelets.

Experimental constructions of self-dual codes

We give a general experimental method generalizing the codes of Carlach and Vervoux (Proceedings of the 13th Applicable Algebra in Engineering Communication and Computing (AAECC 13), Hawaii, USA, 14–19 November 1999, p. 15) to construct self-dual codes. We consider the particular fields GF(2), GF(3), GF(4) (both Euclidean and Hermitian cases), GF(5) and GF(7). We give numerical tables of the best known self-dual codes over these alphabets up to lengths where minimum distances are computable. These tables regularly fill gaps between known codes with good parameters such as the quadratic residue codes, the Pless symmetry codes (J. Combin. Theory Ser. A A12 (1972) 119) or the quadratic double circulant codes (J. Combin. Theory Ser. A 97 (2002) 85). Many new codes with better parameters are constructed: in particular the first extremal ternary self-dual [52,26,15] and [78,39,18] codes and the first binary self-dual [114,57,18] and [116,58,18] codes are constructed. We also give the minimum weight of the Pless symmetry codes of length 84. We also update tables for quadratic residue codes over GF(4), GF(5) and GF(7) and we obtain in particular a [62,31,20] code over GF(5).

Type II Codes over [formula omitted

The natural analogues of Lee weight and the Gray map over F4 are introduced. Self-dual codes for the Euclidean scalar product with Lee weights multiple of 4 are called Type II. They produce Type II binary codes by the Gray map. All extended Q-codes of length a multiple of 4 are Type II. This includes quadratic residue codes attached to a prime p≡3 (mod8), certain double circulant codes, and some affine invariant codes. A general mass formula is derived, a new upper bound for Euclidean self-dual codes over F4 is given, and the first extremal self-dual [92, 46, 16] binary code is built.

On Quadratic Double Circulant Codes over Fields

Abstract In this extended abstract, we give general results about quadratic double circulant codes, which can be seen as a generalization of the Pless Symmetry codes to different fields. In particular new infinite families of self-dual codes over GF(4) and GF(5) are presented. It is proven that the automorphism group of some of these codes contains the group PSL2(q). New codes over GF(4) and GF(5), with better minimum weight than previously known codes, are given.

Extremal double circulant Type II codes over Z 4 and construction of 5-(24, 10, 36) designs

In this paper, we provide a classification of extremal double circulant Type II codes of length 24 over Z 4. It is shown that the sets of supports of the codewords of Hamming weight 10 in certain extremal Type II codes of length 24 form 5-(24, 10, 36) designs. Designs with these parameters were not previously known to exist. A new extremal Type II double circulant code of length 32 is also constructed.

Classification of Extremal Double Circulant Self-Dual Codesof Lengths 64 to 72

Recently extremal double circulant self-dual codes have been classified for lengths n ≤ 62. In this paper, a complete classification of extremal double circulant self-dual codes of lengths 64 to 72 is presented. Almost all of the extremal double circulant singly-even codes given have weight enumerators for which extremal codes were not previously known to exist.

Double circulant self-dual codes over Z/sub 2k/

Recently there has been tremendous interest in self-dual codes over finite rings, specifically the rings Z/sub 2k/. In this paper, we investigate double circulant self-dual codes over Z/sub 2k/ for small k and short lengths. In particular, we give a classification of the extremal double circulant codes. Using invariant theory, a basis for the space of invariants to which the Hamming weight enumerators belong is given for self-dual codes over Z/sub 6/, Z/sub 8/, Z/sub 10/, and Z/sub 12/.

On iterative soft-decision decoding of linear binary block codes and product codes

Iterative decoding methods have gained interest, initiated by the results of the so-called "turbo" codes. The theoretical description of this decoding, however, seems to be difficult. Therefore, we study the iterative decoding of block codes. First, we discuss the iterative decoding algorithms developed by Gallager (1962), Battail et al. (1979), and Hagenauer et al. (1996). Based on their results, we propose a decoding algorithm which only uses parity check vectors of minimum weight. We give the relation of this iterative decoding to one-step majority-logic decoding, and interpret it as gradient optimization. It is shown that the used parity check set defines the region where the iterative decoding decides on a particular codeword. We make plausible that, in almost all cases, the iterative decoding converges to a codeword after some iterations. We derive a computationally efficient implementation using the minimal trellis representing the used parity check set. Simulations illustrate that our algorithm gives results close to soft decision maximum likelihood (SDML) decoding for many code classes like BCH codes. Reed-Muller codes, quadratic residue codes, double circulant codes, and cyclic finite geometry codes. We also present simulation results for product codes and parallel concatenated codes based on block codes.

Type II codes over Z/sub 4/

The conditions satisfied by the weight enumerator of self-dual codes, defined over the ring of integers module four, have been studied by Klemm (1989), then by Conway and Sloane (1993). The MacWilliams (1977) transform determines a group of substitutions, each of which fixes the weight enumerator of a self-dual code. This weight enumerator belongs to the ring of polynomials fixed by the group of substitutions, called the ring R of invariants. Among all of the quaternary self-dual codes, some have the property that all euclidean weights are multiples of 8. These codes are called type II codes by analogy with the binary case. An upper bound on their minimum euclidean weight is given, thereby leading to a natural notion of extremality akin to similar concepts for type II binary codes and type II lattices. The most interesting examples of type II codes are perhaps the extended quaternary quadratic residue codes. This class of codes includes the octacode [8, 4, 6] and the lifted Golay [24, 12, 12]. Other classes of interest comprise a multilevel construction from binary Reed-Muller and lifted double circulant codes.

Weight Enumerators of Double Circulant Codes and New ExtremalSelf-Dual Codes

In this paper it is shown that the weight enumerator of a bordered double circulant self-dual code can be obtained from those of a pure double circulant self-dual code and its shadow through a relationship between bordered and pure double circulant codes. As applications, a restriction on the weight enumerators of some extremal double circulant codes is determined and a uniqueness proof of extremal double circulant self-dual codes of length 46 is given. New extremal singly-even [44,22,8] double circulant codes are constructed. These codes have weight enumerators for which extremal codes were not previously known to exist.