Binary Block Codes Research Articles

Bounds on the Asymptotic Coding Gain of Long Binary Block Codes

For some time it has been known that, for fixed code length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> , binary BCH codes appear to be most efficient when the number of information bits <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> is between <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/4 n</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3/4 n</tex> [1, p. 443], [2, p. 219]. In this correspondence the efficiency of block codes on an binary-quantized additive white Gaussian noise channel is analyzed as a function of the code rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r = k/n</tex> for hard decision decoding. A closed form analytical expression for the upper and lower bounds on block code performance is derived for large code lengths <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> . They show that, for best codes, a relatively broad maximum occurs for rates of approximately 0.4. The performance of the BCH codes is also compared with the bounds.

A Class of Optimum Block Codes in Signal Space

By using convolutional coding coupled with expanded sets of channel signals, Ungerboeck showed that it is possible to obtain coding gains of several decibels at a given data rate without increasing the transmission bandwidth. In this paper, we construct a class of optimum signal space codes using short binary block (as opposed to convolutional) codes. Our results are compared to those obtained by Ungerboeck. A simplified decoding procedure that is asymptotically optimum is also presented.

Reduced search soft-decision minimum-distance decoding for binary block codes

A reduced search soft-decision decoding algorithm for binary block codes is presented. It is based on the weight distribution of the code and the transition probabilities of a binary input Q-ary output channel in sequencing the codeword classes during a table look-up operation. An upper bound is presented for the decoding time improvement factor of the reduced search algorithm compared to exhaustive table look-up decoding.

Probabilistic hard-decision table look-up decoding for binary block codes

A probabilistic hard-decision decoding algorithm based on the weight distribution of binary block codes and the random error distribution in the channel is briefly described. It reduces the number of look-up iterations performed in the conventional exhaustive search table look-up minimum distance decoder.

Frequency Hopping with Multiple Frequency-Shift Keying and Hard Decisions

The performance of frequency-hopping systems with multiple frequency-shift keying and hard decisions is determined for operation against optimal partial-band jamming. A receiver that eliminates the potential advantage of optimally placed jamming tones is proposed. The effects of Reed-Solomon, binary block, and convolutional codes are analyzed. Concatenated codes with hard and soft decisions available to the outer decoder are examined. Continuous-phase frequency-shift keying with limiter-discriminator demodulation is shown to offer potentially improved performance in frequency-hopping systems.

On the Coding Gain of Linear Binary Block Codes

The coding gain of linear binary block codes on the AWGN channel is studied. Antipodal signaling and hard decision demodulation are assumed. Simple asymptotic expressions for the gain at high and low energy to noise spectral density ratios are derived. Monotonicity of the coding gain is also discussed.

Invited paper Line coding for optical fibre systems

Abstract General aspects of coding for optical fibre digital transmission systems are considered in this paper. Simplified mathematical models are developed which reveal the relationship between binary, multilevel and duobinary codes in terms of optical penalty. Various properties of binary block codes are discussed, in particular, how the properties relate to code efficiency.

The enumeration of certain run length sequences

Binary block codes of length n with the property that an arbitrary catenation of codewords will produce a sequence with no runs of either zeros or ones of length greater than k , are considered. The enumeration of maximal codes is determined and the maximal rate such a code may have is found as the logarithm to the base two of the largest real root of a given polynomial.

Coding for radio channels

The conventional remedy to time and/or frequency variability of radio channels is diversity. Redundant coding is a kind of diversity, as each coded symbol can be recovered from other symbols. Only linear binary block codes are considered. Any binary random variable can be represented by its algebraic value,a real number whose sign indicates its most likely value and whose absolute value measures the probability of this value. The algebraic value of a received binary symbol is itself a random variable, whose distribution obeys a particular constraint. The algebraic value associated with the maximum likelihood decision on a binary symbol, given a set of independent received replicas of it, and that associated with the sum modulo 2 of binary random variables are also considered. The symbol-by-symbol decoding is then analysed in the case of threshold decoding, then in the general case. An approximate bound on the decoding error probability for additive Gaussian noise and coherent demodulation is used to assess the advantage of coding when unequalenergy symbols are received, according to a deterministic or a Rayleigh distribution. Simulation results are given for the Hamming (15,11) code. Coding affords a significant advantage provided the channel is good enough, while conventional diversity always provides gain.

Block codes capable of correcting both additive and timing errors

Binary block codes are constructed that are capable of correcting successive timing errors such as the deletion or insertion of s bits (s\leq D) within a single additive burst of errors of length b or less (b=f+g+2D) , where f and D are the design parameters that specify the length of correctable successive timing errors and g is the length of the correctable additive burst. A decoding algorithm is given and the efficiency of these codes is discussed.

Asymptotic performance of optimum bit-by-bit decoding for the white Gaussian channel (Corresp.)

Asymptotic expressions are derived for the probability of bit error for optimum bit-by-bit decoding of a linear binary block code for the white Gaussian noise channel.

Residual error rate of binary linear block codes (Corresp.)

Transposing a computation of Mac Williams we derive an exact expression for, and a straightforward upper bound on, the residual error rate of a binary block code with a "standard" decoder. We also give series expansions for decoding error probability and residual error rates.

A decoding algorithm for binary block codes and&amp;lt;tex&amp;gt;J&amp;lt;/tex&amp;gt;-ary output channels (Corresp.)

A search algorithm is described to decode long binary block codes of any rate for the memoryless binary input <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J</tex> -ary output channel. It can be used directly to perform maximum-likelihood decoding or in a constrained version that gives considerably fewer searches at a small sacrifice in performance. Simulation results are given for a rate-l/2 code of length 128 and minimum Hamming distance 22 on the quantized Gaussian channel.

New families of error-correcting codes generated by modification of other linear binary block codes

Two procedures for modifying linear block error-correction codes are proposed. The first is based on the deletion of x rows and y columns from the parity-check matrix of a given (n, k) code, in such a way that the minimum Hamming distance of the resulting (n−y, k+x−y) code remains equal to that of the original. It is shown that if x is unity, then y may be as low asthe minimum Hamming distance of the (n, n−k) dual of the original (n, k) code. This procedure of deleting rows and columns when applied to the known linear binary block code yields a family of codes, some of which have better rates than those of the best previously known codes of identical Hamming distance and the same number of paritycheck digits. 17 examples of such new codes are derived and included in the paper. It is also shown that apart from appropriate slight modification, the coding and decoding algorithms for this family of modified codes are similar to those of the original code. The second proposed procedure of code modification entails lengthening the original (n, k) linear block code by annexing k′ message digits. If the original code is capable of correcting t random errors or less, then the resulting modified (n+k′,k+k′) code has a rate higher than that of the original (n, k) code. Moreover, its error-correcting capability is such that it will correct t random errors or less if at least one of these occurs in the block of k message digits and s random errors or less, where 1≤s&lt;t, if none of the errors occur in any of the k message digits. Five examples of such codes are given.

On general Gilbert bounds

We define a distance measure for block codes used over memoryless channels and show that it is related to upper and lower bounds on the low-rate error probability in the same way as Hamming distance is for binary block codes used over the binary symmetric channel. We then prove general Gilbert bounds for block codes using this distance measure. Some new relationships between coding theory and rate-distortion theory are presented.

Effective Application of Forward-Acting Error-Control Coding to Multichannel HF Data Modems

The results of a recent study to determine and evaluate effective methods of applying forward-acting error-control-coding techniques to multichannel high-frequency (HF) data modems are presented. A unique feature of this study is a conscious exploitation of the fine-grain bit-error behavior, both in time and in frequency, that is characteristic of multichannel HF data transmission systems. Attention is focused on the ability of four promising coding techniques to combat the characteristic error behavior. These techniques involve the use of bit-interleaved binary block codes, diffuse convolutional codes, block-interleaved two-stage concatenated codes with binary-coded inner stages and nonbinary-coded outer stages, and a class of two-stage binary codes with random-error-correcting inner stages and burst-error correcting outer stages.

Information Rates and Errors Using Decision Feedback

Comparison is made between one-way communication links using binary block codes and a similar link with decision feedback. The known reduction in word error probability is shown. Earlier papers consider only the encoding entropy rate. In this paper the equivocation and rate are computed for single-error detection and single-error correction double-error detection codes with and without feedback.

A family of codes for the correction of substitution and synchronization errors

Binary block codes are constructed that are capable of correcting, in every <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t \geq 3</tex> consecutive words, either one substitution error in each one of, at the most, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-2</tex> words, or one synchronization error (but not both). In order to decode any one word, the decoder needs to wait for the reception of, at the most, the next <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-1</tex> words.

Self-synchronizing codes derived from binary cyclic codes

The sensitivity of a binary block code to loss of synchronism (misplacement of the "commas" separating codewords) can be characterized by a pair of numbers <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[s, \delta]</tex> such that any synchronization slip of s bits or less produces an overlap sequence differing from a legitimate codeword in at least <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta</tex> places. This definition is broader than that of comma freedom of index <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta</tex> , which is included as the special case of s equal to the integer part of half the code block length. For codes having the slip-detecting characteristic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[s, \delta]</tex> there exists the possibility of implementation to restore synchronism during an interval relatively free from bit errors. It is shown that certain error-correcting binary cyclic block codes can be altered to obtain the characteristic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[s, \delta]</tex> by the addition of a fixed binary vector to each codeword prior to transmission. These altered cyclic codes retain the full error-correcting power of the original cyclic codes. An error-detecting/correcting data format providing protection against the acceptance of misframed data is thus obtained without the insertion of special synchronizing sequences into the bit stream.

Optimal error detection codes for completely asymmetric binary channels

The (n/2)-out-of-n code is proved to be the least redundant binary block code which permits the detection of all errors in completely asymmetric channels. It is then proved that the sum code of Berger, Smith, and Freiman is the least redundant of all separable codes of this type. The redundancies of the sum and (n/2)-out-of-n codes are then compared and it is shown that the former is asymptotically twice as redundant as the latter. An efficient method of constructing separable codes which detect up to a given number, but not all, asymmetric errors is included as an appendix.