Trellis complexity and pseudoredundancy of relative two-weight codes

Relative two-weight codes have been studied due to their applications to wiretap channel and secret sharing. It has been shown that these codes form a large family, which includes dual Hamming codes and subcodes of punctured Reed-Muller codes as special instances. This work studies the properties of relative two-weight codes with regard to efficient decoding. More specifically, the trellis complexity, which determines the complexity of Viterbi algorithm based decoding and pseudoredundancy that measures the performance and complexity of linear programming decoding are studied for relative two-weight codes. Separating properties of these codes have been identified and proved first. Based on the results of separating properties, the trellis complexity of binary relative two-weight codes is fully determined. An upper bound on the pseudoredundancy of binary relative two-weight codes is derived.     

Rate 8/9 sliding block distance-enhancing code with stationarydetector

A new distance-enhancing code for partial-response magnetic recording channels eliminates most frequent errors, while keeping the two-step code trellis time invariant. Recently, published trellis codes either have lower code rates or result in time-varying trellises with a period of nine, thus requiring a higher complexity of detectors and code synchronization. The new code introduces dependency between code words in order to achieve the same coding constraints as the 8/9 time-varying maximum transition runlength (TMTR) code, with the same code rate, but resulting in a trellis that has a period of 2. This code has been applied to the E²PR4 and a 32-state generalized partial response (GPR) ISI target. The resulting two-step trellises have 14 and 28 states, respectively. Coding gain is demonstrated for both targets in additive white Gaussian noise     

平坦快衰落条件下天线数较多时空时格码的设计

空时格码技术是近年来无线通信领域的一个研究热点，空时格码的设计准则问题一直是该技术的一个难点。在平坦快衰落条件下且天线数目较多时，基于最小平方欧式距离最大化的空时格码设计准则虽然优于经典的距离－积准则，但仍不完善。从平方距离分布与平方距离和方－方和比分布的角度对此进行了改进，得到改进的准则以及性能为目前最优的新码。还对快慢2种衰落条件下码性能差异的原因给出了新的解释，并将改进的码设计准则应用于智贪码的设计。     

迫零对Turbo码的性能影响

从距离谱的角度对迫零在Turbo码中的作用进行了理论分析。通过仿真，研究了迫零对Turbo码性能的影响。仿真结果表明，Turbo码编码器迫零处理有助于改善Turbo码的性能。     

Super-orthogonal space-time-frequency trellis coded OFDM

Super-orthogonal space-time trellis codes (SOSTTCs) given in the literature are full rate and provide full spatial diversity with high coding gain in narrowband quasistatic fading channels. The high number of parallel transitions in their trellis structure restricts their performance in wideband channels where the code performance suffers from multipath. Code design criteria are derived and a computer search based method is proposed to design full-rate optimised SOSTTCs exploiting full spatial and multipath diversity for multiple-input multiple-output orthogonal frequency division multiplexing systems. The proposed codes have codewords defined in space, time, and frequency. We evaluate the codeword error rates of the new 16, 32 and 64-state super-orthogonal space-time-frequency trellis codes for quadrature phase shift keying by computer simulation and show that they provide significant error performance improvement compared to their counterparts with equivalent delay length     

Trellis coded modulation based distributed channel coding

A novel technique is described for distributed channel coding in wireless communication networks. The protocol is based on a block fading model of the multi-user uplink channel and on adapting multidimensional trellis coded modulation techniques to a coding theorem derived for the block fading channel. The coded modulation (CM) protocol is designed to optimise code performance, especially at high spectral efficiencies. The CM protocol is very simple to implement at the cooperating users. Complexity is completely transferred to the destination. The latency of the protocol is only one symbol. The CM protocol achieves full diversity order equal to the number of cooperating users and maximises coding gains by designing the code for the specific modulation used. We derive analytical results and present simulation results showing the benefits of CM protocol over other comparable schemes.     

Bandwidth-efficient code design for coordinate interleaved coded cooperation

A two-user coordinate interleaved coded cooperation scheme is proposed for quasi-static Rayleigh fading channels, where cooperative and modulation diversity techniques are properly combined to take their full advantage. Two selective cooperation schemes are considered related to whether users know the cooperation case or not. Pairwise error probability (PEP) analysis is performed for all cooperation cases and code design criteria are derived from the PEP upper bounds. Bandwidth-efficient 4-, 8- and 16-state rate 2/4 cooperative quadrature phase shift keying (QPSK) trellis codes are obtained based on these criteria by means of exhaustive computer search. The error performances of the new codes evaluated by computer simulations show their superiority compared to the corresponding best space-time codes used in cooperation with coordinate interleaving. The simulation results are supported by an upper bound on the bit error probability developed using union bounding technique.     

On minimum distance bounds for abelian codes

This paper is an exposition of two methods of formulating a lower bound for the minimum distance of a code which is an ideal in an abelian group ring. The first, a generalization of the cyclic BCH (Bose-Chaudhuri-Hoquenghem) bound, was proposed by Camion [2]. The second method, presented by Jensen [4], allows the application of the BCH bound or any of its improvements by viewing an abelian code as a direct sum of concatenations of cyclic codes. This second method avoids the mathematical analysis required for a direct generalization of a cyclic bound to the abelian case. It can produce a lower bound that improves the generalized BCH bound. We present simple algorithms for 1) deriving the generalized BCH bound for an abelian code 2) determining direct sum decompositions of an abelian code to concatenated codes and 3) deriving a bound on an abelian code, viewed as a direct sum of concatenated codes, by applying the cyclic BCH bound to the inner and outer code of each concatenation. Finally, we point out the applicability of these methods to codes that are not ideals in abelian group rings.     

Maximum Distance Separable Convolutional Codes

A maximum distance separable (MDS) block code is a linear code whose distance is maximal among all linear block codes of rate k/n. It is well known that MDS block codes do exist if the field size is more than n. In this paper we generalize this concept to the class of convolutional codes of a fixed rate k/n and a fixed code degree δ. In order to achieve this result we will introduce a natural upper bound for the free distance generalizing the Singleton bound. The main result of the paper shows that this upper bound can be achieved in all cases if one allows sufficiently many field elements. Received: December 10, 1998; revised version: May 14, 1999     

The Minimal Generator Matrix of a Vector Space

The atomic vectors of a finitely generated vector space C over a field F are characterized for C a subspace of the product vector space ? = ∏ _{i =1} ⁿ ? _i over F. For finite fields, the minimal trellis diagram for mixed-codes is determined, and this provides the L-section minimal trellis diagram for linear codes. As an example, an extremely simple yet comprehensive analysis of the trellis structure of Reed-Muller codes is given. In particular, a trellis oriented generator matrix for the 2 ^l -section minimal trellis diagram of a Reed-Muller code is presented. Received: February 27, 1997; revised version: May 6, 1999     

Weight distribution of turbo codes with convolutional interleavers

A simple algorithm for the weight calculation of turbo codes with convolutional interleavers is presented. For codes with short interleaver lengths, the weight distributions are computed using conventionally proposed methods and then utilised together with the interleaver properties to determine the weight specifications for the code with a longer desired length. Based on the calculated weights, a new upper bound for the code is computed. It agrees with simulation results of the code performance in the error-floor region     

Generalized quasi-cyclic codes: structural properties and code construction

Generalized quasi-cyclic (GQC) codes are defined by generator matrices comprised of circulant matrices of lengths not necessarily identical. A decomposition of these codes is given by using the Chinese reminder theorem. The focus is to characterize ρ-generator GQC codes in details. A good lower bound on the minimum distance of such a code in terms of the minimum distance of the constituent codes is given. Construction methods are given and a set of GQC codes is provided that from minimum distance perspective are optimal codes among the known linear codes having the same length and dimension.     

The synthesis of magnetic recording trellis codes with good hamming distance properties

In this paper we present, by means of an example, a systematic procedure to synthesize combined modulation/error correcting trellis codes, suitable for Viterbi decoding. This synthesis is based on firstly selecting a suitable linear convolutional code, secondly by analysing the state system of this code to determine the important Hamming distance building properties, and finally by mapping a code with the desired restrictions on its sequences onto this state system. As an example we develop a R = 3/6 dc free (b,l,c) = (0,3,2) code withd_{min} = 4. This code improves on the best codes in [1]. Codes havingb geq 1, and which will thus be more suitable for magnetic recording, can also be synthesized following the proposed procedure.     

DFT Domain Characterization of Quasi-Cyclic Codes

 The transform domain characterization of linear cyclic codes over finite fields using Discrete Fourier Transform (DFT) over an appropriate extension field is well known. In this paper, we extend this transform domain characterization for linear quasi-cyclic codes over finite fields. We show how one can derive a lower bound on the minimum Hamming distance of a quasi-cyclic code and decode the code upto that minimum Hamming distance using this characterization. Received: January 17, 2002; revised version: November 30, 2002 RID="*" ID="*" This work was partly supported by CSIR, India, through Research Grant (22(0298)/99/EMR-II) to B. S. Rajan Keywords: Quasi-cyclic codes, Discrete Fourier transform, Cyclotomic cosets. Part of this work was presented in ICCCD 2000, Kharagpur, India and ISIT 2001, Washington D.C., USA     

Maximum-girth slope-based quasi-cyclic (2, k ⩾ 5) low-density parity-check codes

A class of maximum-girth geometrically structured regular (n, 2, k ⩾ 5) (column-weight 2 and rowweight k) quasi-cyclic low-density parity-check (LDPC) codes is presented. The method is based on cylinder graphs and the slope concept. It is shown that the maximum girth achieved by these codes is 12. A lowcomplexity algorithm producing all such maximum-girth LDPC codes is given. The shortest constructed code has a length of 105. The minimum length n of a regular (2, k) LDPC code with girth g ? 12 determined by the Gallager bound has been achieved by the constructed codes. From the perspective of performance these codes outperform the column-weight 2 LDPC codes constructed by the previously reported methods. These codes can be encoded using an erasure decoding process.     

Fast decoding of non-binary first order Reed-Muller codes

A minimum distance decoding algorithm for non-binary first order Reed-Muller codes is described. Suggested decoding is based on a generalization of the fast Hadamard transform to the non-binary case. We also propose a fast decoding algorithm for non-binary first order Reed-Muller codes with complexity proportional to the length of the code. This algorithm provides decoding within the limits guaranteed by the minimum distance of the code.Partly supported by the Guastallo Fellowship. This work was presented in part at the 9th International Symposium Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, New Orleans, USA, October 1991     

码率嵌入式高码率差分空时编码

为了提高传输码率,将具有高码率的全分集全码率(FDFR)空时码嵌入到低码率的传统差分空时编码中,提出了一种码率嵌入式的差分空时编码方法,该编码方案具有编码增益高,误码性能好等优点.仿真结果表明,与传统的差分空时编码方案(酉差分空时编码和差分空时分组编码)相比,新编码方法具有很好的误码性能,特别是在高传输码率和接收天线较多情况下,其误码性能优势越明显.     

Concatenated bit-interleaved coded modulation and orthogonal space-time block codes over fading channels

The authors analyse concatenated bit-interleaved coded modulation and orthogonal space-time block codes (OSTBC) over fading channels in the absence and presence of channel state information (CSI) in receiver. The authors derive analytical expressions for bit and frame error probabilities based on which corresponding designing rules are proposed. The analytical results are for arbitrary rate of constituent STBC and arbitrary convolutional code, and for CSI-aware receiver is for any number of transmit and receive antennas. Simulation results are presented to confirm the validity of the proposed designing rules. Moreover, the simulation results show that the proposed system outperforms concatenated trellis coded modulation and OSTBC.     

Optimal code rates for the Lorentzian channel: Shannon codes and LDPC codes

We take an information-theoretic approach to obtaining optimal code rates for error-control codes on a magnetic storage channel approximated by the Lorentzian channel. Code rate optimality is in the sense of maximizing the information-theoretic user density along a track. To arrive at such results, we compute the achievable information rates for the Lorentzian channel as a function of signal-to-noise ratio and channel density, and then use these information rate calculations to obtain optimal code rates and maximal linear user densities. We call such (hypothetical) optimal codes "Shannon codes." We then examine optimal code rates on a Lorentzian channel assuming low-density parity-check (LDPC) codes instead of Shannon codes. We employ as our tool extrinsic information transfer (EXIT) charts, which provide a simple way of determining the capacity limit (or decoding threshold) for an LDPC code. We demonstrate that the optimal rates for LDPC codes coincide with those of Shannon codes and, more important, that LDPC codes are essentially capacity-achieving codes on the Lorentzian channel. Finally, we use the above results to estimate the optimal bit-aspect ratio, where optimality is in the sense of maximizing areal density.     

Frequency-hopping optical orthogonal codes with arbitrary time-blank patterns

We investigate a general scheme of frequency-hopping optical orthogonal codes with a specified distance between adjacent frequency symbols and propose a novel code that allows time blanks between adjacent frequency symbols in code sequences. A time blank represents the absence of frequency symbols in code sequences and makes no interference with frequency components. The insertion of time-blank patterns can provide ample scope to generate much more code sequences than the conventional codes lacking in time-blank patterns, and we show this by constructing an algorithm to generate the proposed code. The performance analysis demonstrates that its performance is superior to that of the conventional codes in terms of the bit error rate. We also derive the upper bound on the proposed code set.