A comparison of trellis modules for binary convolutional codes

A convolutional code can be represented by the conventional trellis module or the “minimal” trellis module based on the BCJR trellis for block codes. For many convolutional codes, the trellis complexity (TC) of the minimal module is significantly less than the TC of the conventional module. An alternative representation, consisting of an extended BCJR trellis module and a pruned version of the conventional module, was previously introduced. We prove that the overall TC of the two new modules is less than the TC of the conventional module for infinitely many codes. Furthermore, we show that the overall TC of the new modules is smaller than the TC of the minimal module for many codes considered in the literature     

The weight distributions of some binary quadratic residue codes

The weight distributions of binary quadratic residue codes C can be computed from the weight distribution of a subset of C containing one-fourth (resp., one-eighth) of the codewords in C when the length of the code is congruent to 1 (resp., -1) modulo 8. An algorithm to determine the weight distributions of binary cyclic codes is given. As a consequence, the weight distributions of (73,37,13), (89,45,17), and (97,49,15) quadratic residue codes are determined precisely.     

Universal decoding algorithm for binary quadratic residue codes using syndrome-weight determination

A novel decoding scheme, called syndrome-weight determination, was proposed by Chang et al. in 2008 for the Golay code, or the (23, 12, 7) quadratic residue code. This method is not only very simple in principle but also suitable for parallel hardware design. Presented is a modified version for any binary quadratic residue codes which has been developed. Because of its regular property, the proposed decoder is suitable for both software design and hardware development.     

A Gilbert bound for periodic binary convolutional codes

A Gilbert bound for periodic binary convolutional (PBC) codes is established. This bound shows that regardless of previous decoding decisions any fraction of errors less thanalpha/2can be corrected in a constraint length by some PBC code if the constraint length is sufficiently large andR, the code rate, is less than[1 - H(alpha)]/2, 0 leq alpha < frac{1}{2}.     

On the performance of rate 1/2 convolutional codes with QPSK onRician fading channels

Consideration is given to the bit error probability performance of rate 1/2 convolutional codes in conjunction with quaternary phase shift keying (QPSK) modulation and maximum-likelihood Viterbi decoding on fully interleaved Rician fading channels. Applying the generating function union bounding approach, an asymptotically tight analytic upper bound on the bit error probability performance is developed under the assumption of using the Viterbi decoder with perfect fading amplitude measurement. Bit error probability performance of constraint length K=3-7 codes with QPSK is numerically evaluated using the developed bound. Tightness of the bound is examined by means of computer simulation. The influence of perfect amplitude measurement on the performance of the Viterbi decoder is observed. A performance comparison with rate 1/2 codes with binary phase shift keying (BPSK) is provided     

'Diffuse' threshold decodable rate 1/2 convolutional codes

This paper shows the existence (by construction) of ratefrac{1}{2}convolutional codes that correct bursts of lengthBortrandom errors. The codes have memory length and guard space requirements that are asymptotically4Band minimal effective length. Kohlenberg introduced the term "diffuse code" for his two random-error or single-burst error-correcting codes with minimal effective length and both a memory and guard-space requirement of3B + 1. To this point, there does not seem to be any procedure for finding codes with minimal effective length and guard space requirements asymptotically3B.     

A maximum-likelihood soft-decision sequential decoding algorithmfor binary convolutional codes

We present a trellis-based maximum-likelihood soft-decision sequential decoding algorithm (MLSDA) for binary convolutional codes. Simulation results show that, for (2, 1, 6) and (2, 1, 16) codes antipodally transmitted over the AWGN channel, the average computational effort required by the algorithm is several orders of magnitude less than that of the Viterbi algorithm. Also shown via simulations upon the same system models is that, under moderate SNR, the algorithm is about four times faster than the conventional sequential decoding algorithm (i.e., stack algorithm with Fano metric) having comparable bit-error probability     

Variable-complexity trellis decoding of binary convolutional codes

Considers trellis decoding of convolutional codes with selectable effort, as measured by decoder complexity. Decoding is described for single parent codes with a variety of complexities, with performance “near” that of the optimal fixed receiver complexity coding system. Effective free distance is examined. Criteria are proposed for ranking parent codes, and some codes found to be best according to the criteria are tabulated, Several codes with effective free distance better than the best code of comparable complexity were found. Asymptotic (high SNR) performance analysis and error propagation are discussed. Simulation results are also provided     

一种低复杂度双二元卷积Turbo码译码算法

双二元卷积Turbo码（ DB CTC)的非二进制编码使得译码复杂度增加,限制了其在某些实际通信工程中的使用。在最大后验概率（ MAP)译码算法的基础上,提出了一种优化算法,将译码的存储量和计算量降为原来的1/4。仿真结果表明：在不同编码长度和码率的情况下,优化算法与原算法性能相当;在误码率为10-5的条件下,两者的Eb/N0差异同样不大于0.1 dB。     

The automorphism groups of the generalized quadratic residue codes

We give the full automorphism groups as groups of semiaffine transformations, of the extended generalized quadratic residue codes. We also present a proof of the Gleason-Prange theorem for the extended generalized quadratic residue codes that relies only on their definition and elementary theory of linear characters     

Iterative decoding of binary block and convolutional codes

Iterative decoding of two-dimensional systematic convolutional codes has been termed “turbo” (de)coding. Using log-likelihood algebra, we show that any decoder can be used which accepts soft inputs-including a priori values-and delivers soft outputs that can be split into three terms: the soft channel and a priori inputs, and the extrinsic value. The extrinsic value is used as an a priori value for the next iteration. Decoding algorithms in the log-likelihood domain are given not only for convolutional codes but also for any linear binary systematic block code. The iteration is controlled by a stop criterion derived from cross entropy, which results in a minimal number of iterations. Optimal and suboptimal decoders with reduced complexity are presented. Simulation results show that very simple component codes are sufficient, block codes are appropriate for high rates and convolutional codes for lower rates less than 2/3. Any combination of block and convolutional component codes is possible. Several interleaving techniques are described. At a bit error rate (BER) of 10^-4 the performance is slightly above or around the bounds given by the cutoff rate for reasonably simple block/convolutional component codes, interleaver sizes less than 1000 and for three to six iterations     

Improved rate 1/4 convolutional codes

The performance of parallel architectures for generating rate 1/4 high constraint length convolutional codes is examined via information weight distributions. It is found that these codes offer a significant decoding advantage, while having a comparable bit error rate (BER) performance to classical codes     

A double circulant presentation for quadratic residue codes (Corresp.)

Forp eqiv pm 1 pmod{8}there are two binary codes,Q(p)andN(p), each an extended quadratic residue code of lengthp+1and dimension(p+1)/2. The existence of double circulant generator matrices for these codes is investigated. A possibly infinite family of primespis presented for whichQ(p)andN(p)must have double circulant generator matrices. Two counterexamples prove the construction is not always possible.     

On the binary quadratic residue system with noncoprime moduli

The residue number system (RNS) appropriate for implementing fast digital signal processors since it can support parallel, carry-free, high-speed arithmetic. A development in residue arithmetic is the quadratic residue number system (QRNS), which can perform complex multiplications with only two integer multiplications instead of four. An RNS/QRNS is defined by a set of relatively prime integers, called the moduli set, where the choice of this set is one of the most important design considerations for RNS/QRNS systems. In order to maintain simple QRNS arithmetic, moduli sets with numbers of forms 2ⁿ+1 (n is even) have been considered. An efficient such set is the three-moduli set (2^2k-2+1.2^2k+1.2^2k+2+1) for odd k. However, if large dynamic ranges are desirable, QRNS systems with more than three relatively prime moduli must be considered. It is shown that if a QRNS set consists of more than four relatively prime moduli of forms 2ⁿ+1, the moduli selection process becomes inflexible and the arithmetic gets very unbalanced. The above problem can be solved if nonrelatively prime moduli are used. New multimoduli QRNS systems are presented that are based on nonrelatively prime moduli of forms 2ⁿ +1 (n even). The new systems allow flexible moduli selection process, very balanced arithmetic, and are appropriate for large dynamic ranges. For a given dynamic range, these new systems exhibit better speed performance than that of the three-moduli QRNS system     

No binary quadratic residue code of length 8m-1 is quasi-perfect

The class of binary quadratic residue (QR) codes of length n=8m-1 contains two perfect codes. These are the (7,4,3) Hamming code and the (23,12,7) Golay code. However, it is proved in the present paper that there are no quasi-perfect QR codes of length 8m-1. Finally, this result is generalized to all binary self-dual codes of length N>72     

Some rate 1/3 and 1/4 binary convolutional codes with an optimum distance profile (Corresp.)

A tabulation of binary systematic convolutional codes with an optimum distance profile for rates1/3and1/4is given. A number of short rate1/3binary nonsystematic convolutional codes are listed. These latter codes are simultaneously optimal for the following distance measures: distance profile, minimum distance, and free distance; they appear attractive for use with Viterbi decoders. Comparisons with previously known codes are made.     

Optimal free distance convolutional codes for rates 1/2, 1/3 and1/4

Optimum distance profile (ODP) codes are good codes for convolutional codes. These codes can be determined using sieve techniques and the improved Cedervall-Johannesson algorithm. The authors present extensive lists of OFD codes containing rates R=1/2, 1/3 and 1/4     

Evaluating the performance of convolutional codes over block fadingchannels

This correspondence considers union upper bound techniques for error control codes with limited interleaving over block fading Rician channels. A modified bounding technique is presented that relies on limiting the conditional union bound before averaging over the fading process. This technique, although analytically not very attractive, provides tight and hence useful numerical results     

On the minimum distance of some quadratic residue codes (Corresp.)

We find the minimum distances of the binary(113, 57), and ternary(37, 19), (61, 31), (71, 36), and(73, 37)quadratic residue codes and the corresponding extended codes. These distances are15, 10, 11, 17, and17, respectively, for the nonextended codes and are increased by one for the respective extended codes. We also characterize the minimum weight codewords for the(113, 57)binary code and its extended counterpart.     

Short binary convolutional codes with maximal free distance for rates 2/3 and 3/4 (Corresp.)

A search procedure is developed to find good short binary(N,N - 1)convolutional codes. It uses simple rules to discard from the complete ensemble of codes a large fraction whose free distanced_{free}either cannot achieve the maximum value or is equal tod_{free}of some code in the remaining set. Farther, the search among the remaining codes is started in a subset in which we expect the possibility of finding codes with large values ofd_{free}to be good. A number of short, optimum (in the sense of maximizingd_{free}), rate-2/3 and 3/4 codes found by the search procedure are listed.