DC-free error-correcting convolutional codes [comment and reply]

In a recent Letter, Deng et. al. (see ibid., vol.29, no.22, pp.1910-1911, 1993) reported a new class of convolutional codes which have both DC-free and error-correcting properties. The presented codes have a null at DC. However, there are some flaws in these codes. When encoding the information sequence with controlling the polarity of disparity d(t) of the codeword, the decoded sequence is different from the information sequence despite the error free condition. We show this by an example. A reply by Deng and Li is included     

Constructions of error-correcting DC-free block codes

Two DC-free codes are presented with distance 2d, b ⩾1 length 2n+2r(d-1) for d⩽3 and length 2n+2r(d-1)(2d -1) for d>3, where r is the least integer ⩾log₂ (2n+1). For the first code l=4, c=2, and the asymptotic rate of this code is 0.7925. For the second code l=6, c=3, and the asymptotic rate of this code is 0.8858. Asymptotically, these rates achieve the channel capacity. For small values of n these codes do not achieve the best rate. As an example of codes of short length with good rate, the author presents a (30, 10, 6, 4) DC-free block code with 2²¹ codewords. A construction is presented for which from a given code C ₁ of length n, even weight, and distance 4, the author obtains a (4n, l, c, 4) DC-free block code C₂, where l is 4, 5 or 6, and c is not greater than n+1 (but usually significantly smaller). The codes obtained by this method have good rates for small lengths. The encoding and decoding procedures for all the codes are discussed     

A class of DC-free subcodes of convolutional codes

We describe a class of DC-free subcodes of convolutional codes that satisfy certain runlength constraints and that also possess error-correcting capability. The running disparity and the maximum runlength of these codes are bounded by quantities that are independent of the free distance. They can be decoded using a Viterbi decoder for the underlying convolutional code     

DC-free binary convolutional coding

A novel DC-free binary convolutional coding scheme is presented. The proposed scheme achieves the DC-free coding and error-correcting capability simultaneously. The scheme has a simple cascaded structure of the running digital sum (RDS) control encoder and the conventional convolutional encoder. A given sequence becomes DC-free if and only if the absolute RDS value of the sequence is bounded by a constant for any time instant. The RDS control encoder generates a sequence which gives the convolutional-coded sequence with a bounded RDS value. The structure allows us to exploit efficient soft-decision decoding which attains additional coding gains compared with hard-decision decoding over an additive white Gaussian noise (AWGN) channel. Bounds on the RDS value are explicitly established for the proposed scheme. By using the bounds, we have performed computer searches for finding good RDS control encoders. The proposed scheme provides wide varieties of reasonable tradeoffs between the coding gain, the RDS constraint, and decoding complexity. For example, a 64-state DC-free coding scheme with the overall rate 6/16 and the minimum free distance 10 has been obtained. This scheme satisfies a bounded RDS constraint (from -18 to +18) and it yields a considerably high asymptotic coding gain (over an AWGN channel) of 5.7 dB     

DC-free coset codes

A class of block coset codes with disparity and run-length constraints are studied. They are particularly well suited for high-speed optical fiber links and similar channels, where DC-free pulse formats, channel error control, and low-complexity encoder-decoder implementations are required. The codes are derived by partitioning linear block codes. The encoder and decoder structures are the same as those of linear block codes with only slight modifications. A special class of DC-free coset block codes are derived from BCH codes with specified bounds on minimum distance, disparity, and run length. The codes have low disparity levels (a small running digital sum) and good error-correcting capabilities     

DC-free error-control block codes

DC-free codes and error-control (EC) codes are widely used in digital transmission and storage systems. To improve system performance in terms of code rate, bit-error rate (BER), and low-frequency suppression, and to provide a flexible tradeoff between these parameters, this paper introduces a new class of codes with both dc-control and EC capability. The new codes integrate dc-free encoding and EC encoding, and are decoded by first applying standard EC decoding techniques prior to dc-free decoding, thereby avoiding the drawbacks that arise when dc-free decoding precedes EC decoding. The dc-free code property is introduced into standard EC codes through multimode coding techniques, at the cost of minor loss in BER performance on the additive white Gaussian noise channel, and some increase in implementation complexity, particularly at the encoder. This paper demonstrates that a wide variety of EC block codes can be integrated into this dc-free coding structure, including binary cyclic codes, binary primitive BCH codes, Reed-Solomon codes, Reed-Muller codes, and some capacity-approaching EC block codes, such as low-density parity-check codes and product codes with iterative decoding. Performance of the new dc-free EC block codes is presented.     

Superimposed codes based on punctured convolutional codes

Punctured convolutional codes of rates k₁/n and k₂ /n are applied to |u|u+v construction, and then a superimposed code of rate (k₁+k₂)/(2n) is constructed. A suboptimal decoding procedure is presented for the superimposed codes, and it reduces the decoding complexity as compared with maximum likelihood decoding for the known convolutional codes     

Good error-correcting codes based on very sparse matrices

We study two families of error-correcting codes defined in terms of very sparse matrices. “MN” (MacKay-Neal (1995)) codes are recently invented, and “Gallager codes” were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties. The decoding of both codes can be tackled with a practical sum-product algorithm. We prove that these codes are “very good”, in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit. This result holds not only for the binary-symmetric channel but also for any channel with symmetric stationary ergodic noise. We give experimental results for binary-symmetric channels and Gaussian channels demonstrating that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed, the performance of Gallager codes is almost as close to the Shannon limit as that of turbo codes     

Analog error-correcting codes based on chaotic dynamical systems

The properties of chaotic dynamical systems make them useful for channel coding in a variety of practical communication applications. To illustrate this, a novel analog code based on tent map dynamics and having a fast decoding algorithm is developed for use on unknown, multiple, and time-varying signal-to-noise ratio (SNR) channels. This code is shown to be an attractive alternative to both digital codes and linear modulation in such scenarios. Several properties and interpretations of the codes are developed, along with some methods for their optimization     

Short error-correcting codes

Theoretical bounds on the error-correcting capabilities of binary codes can be found for very long codes in terms of k/n and t/n. It is pointed out that these bounds are only valid for very long codes, and that the relationship between k/n and t/n is not a sufficient criterion for the practical value of short codes.     

基于混沌的纠错编码设计和实现

本文在扩展卡尔曼滤波、混沌信号模拟调制,以及数字纠错编码的码距离度量等思想的基础上,设计和实现了一套基于混沌的模拟纠错编码/解码算法.仿真结果表明在低信噪比情况下,该算法有较强的纠错能力,优于传统的分组交织纠错编码.     

A further note on DC-free coset codes

The procedure introduced by R.H. Deng and M.A. Herro (see ibid., vol.IT-34, p.786-92, 1988) for identifying DC-free codes is considered further. Certain of the detailed statements relating to disparity and runlength are corrected and/or refined. While this is considered important for a proper interpretation of the codes and encoding procedure it is emphasized that it in no way detracts from the usefulness of the general algorithmic approach for producing these codes     

Comma-free error-correcting codes

A method is shown by which it is possible to establish the existence (or nonexistence) of the comma-free properties of any group code from a simple observation of its null-space. Using this technique it is then demonstrated that all(n, k){em cyclic} group error-correcting code dictionaries can be made comma-free (without adding further redundancy or altering their error-correcting properties) if, and only if,k leq (n - 1)/2. Ifkis larger than(n - 1)/2it is still possible to detect bit gains or losses of up ton-k-1code bits.     

Binoid error-correcting codes

The purpose of this paper is to develop more general techniques for the synthesis of error-correcting codes that dispense with the requirement that coding elements and operations must be associated with finite fields. This approach permits an extension of the class of coded messages and coding operations and leads in certain cases to a reduction of implementation complexity.     

Symbol-by-symbol reception of signals corresponding to high-rate convolutional codes and turbo codes based on these convolutional codes

Computational procedures for symbol-by-symbol reception of signal ensembles corresponding to binary high-rate convolutional codes and to turbo codes formed by these convolutional codes are described. It is shown that the developed procedures are based on an optimum symbol-by-symbol reception algorithm that uses the fast Walsh-Hadamard transform algorithm.     

Cyclically permutable error-correcting codes

In an asynchronous multiplex system a single band of radio frequencies carries many simultaneous conversations, each of which uses a distinguishable train of pulses as a "carrier." To design a good list of allowed pulse trains one is led to a peculiar kind of error-correcting code problem. The encoding sought uses binaryN-tuples (code words) in which most of the digits are zero. In addition the Hamming distances from any given code word to all cyclic permutations of all other code words must exceed a prescribed minimum value. The code words found herein may be generated simply using a shift register.     

Accumulate codes based on 1+D convolutional outer codes

A new construction of good, easily encodable, and soft-decodable codes is proposed in this paper. The construction is based on serially concatenating several simple 1+D convolutional codes as the outer code, and a rate-1 1/(1+D) accumulate code as the inner code. These codes have very low encoding complexity and require only one shift-forward register for each encoding branch. The input-output weight enumerators of these codes are also derived. Divsalar?s simple bound technique is applied to analyze the bit error rate performance, and to assess the minimal required signal-to-noise ratio (SNR) for these codes to achieve reliable communication under AWGN channel. Simulation results show that the proposed codes can provide good performance under iterative decoding.     

On error-correcting balanced codes

Results are presented on families of balanced binary error-correcting codes that extend those in the literature. The idea is to consider balanced blocks as symbols over an alphabet and to construct error-correcting codes over that alphabet. Encoding and decoding procedures are presented. Several improvements to the general construction are discussed     

Generalized algorithm for design of DC-free codes based onmultilevel partition chain

A generalized algorithm for design of DC-free codes based on the multilevel partition chain is presented. The multilevel partition chain is formed from the code partition chain of Reed-Muller codes. Some parameters such as the maximum runlength and running digital sum (RDS) of DC-free codes are also obtained. The results show that the DC-free codes obtained by the presented design algorithm have good coding parameters     

Low-frequency performance of guided scrambling DC-free codes

We analyze the effect of coding parameters on low-frequency suppression in guided scrambling (GS) dc-free codes based on computer simulation. Simple expressions to accurately estimate the sum variance and low-frequency spectrum-weight of these codes are developed. Systematic design of GS dc-free codes is discussed.