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Chapman & Hall/CRC eBooks

Binary linear codes

Juergen Bierbrauer

Published Oct 3, 2018• 0 citations
Publisher

Abstract

Binary linear codes are particularly handy for block coding. This chapter describes how block coding works: the incoming data is divided into consecutive blocks of length k . In block coding the length of bitstrings is increased from k to n . The chapter compares the error probabilities with and without coding in a special case, based on the binary symmetric channel. It examines how to describe binary linear codes using either a generator matrix or a check matrix. It is natural to concentrate on the rows of these matrices. However, it will turn out very profitable to change the point of view and read these matrices column wise. The notion of an orthogonal array originated not in information theory but in statistics. It plays a central role in the design of experiments and has many more applications in theoretical computer science and cryptology.

DOI
10.1201/9781482296372-2

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Binary linear codes are particularly handy for block coding. This chapter describes how block coding works: the incoming data is divided into consecutive blocks of length k . In block coding the length of bitstrings is increased from k to n . The chapter compares the error probabilities with and without coding in a special case, based on the binary symmetric channel. It examines how to describe binary linear codes using either a generator matrix or a check matrix. It is natural to concentrate on the rows of these matrices. However, it will turn out very profitable to change the point of view and read these matrices column wise. The notion of an orthogonal array originated not in information theory but in statistics. It plays a central role in the design of experiments and has many more applications in theoretical computer science and cryptology.
A binary code is linear if it is closed under addition. It is a basic fact from linear algebra that each linear code has a basis. Linear algebra applies to arbitrary fields. There is a general tendency to restrict attention to linear codes. One reason is that these are much easier to describe and to work with than codes in general. For a binary linear code, the minimum distance equals the minimum of the weights of nonzero codewords. Binary linear codes are particularly handy for block coding. In block coding the length of bitstrings is increased from k to n. That means that k/n of the bits sent through the channel represent pure information, whereas the rest is redundancy that we have cleverly introduced for the purposes of error correction.
The performance of optimum linear block codes has been examined by computer simluation, and the error distribution at the decoder output, the block-error probability and the bit-error probability have been determined for the codes when they are used on a binary symmetric channel with random errors. A table showing the distribution of errors in the decoder output is presented, and it is shown how the table can be used to determine exact values for the bit-error probability and the block-error probability. The exact values of bit-error probability computed from the table are compared with values obtained using a known theoretical approximation to the bit-error probability, and some conclusions are drawn as to the general usefulness of the theoretical approximation. Also, block-error probabilities are computed from the information presented in the table and these values are compared with the estimates of block-error probability obtained from minimumdistance considerations.
A low-density parity-check code is a code specified by a parity-check matrix with the following properties: each column contains a small fixed numberj \geq 3of l's and each row contains a small fixed numberk > jof l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixedj. When used with maximum likelihood decoding on a sufficiently quiet binary-input symmetric channel, the typical probability of decoding error decreases exponentially with block length for a fixed rate and fixedj. A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described. Both the equipment complexity and the data-handling capacity in bits per second of this decoder increase approximately linearly with block length. Forj > 3and a sufficiently low rate, the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length. Some experimental results show that the actual probability of decoding error is much smaller than this theoretical bound.
We consider the problem of decoding a linear block code used over a binary symmetric channel with crossover probability p when the goal of the decoding is to minimize the probability of an information bit error. In general, there will be several different decoding strategies, each corresponding to the optimal rule over some range p/spl isin/[p/sub i-1/, p/sub i/]. Each of these strategies can be implemented by standard array with the appropriate choice of coset leaders that are not in general of minimum weight. We give optimal rules for selecting coset leaders in the first region [0, p/sub 1/] and the region [p/sub max/, 1/2] for any linear mapping between information bits and codewords. The determination of the optimal rule has been an open problem for small p. The rule for p>p/sub max/ is important because for many codes, when p is fixed the optimal rule for any coset corresponds to either the optimal rule for small p, or the optimal rule for p near 1/2. The optimal decoding strategy varies with the generator matrix used in general, but it is demonstrated that the optimal generator matrix is separable when d/sup /spl perp//>2. The results are shown to extend in a natural way to the case where the goal is to minimize the probability that arbitrary collections of information bits are in error. We examine some specific codes, and compare the performance under optimal decoding to the performance of previously suggested strategies.>
A coset representation of the bit error probabilities for binary linear block codes and arbitrary symmetric memoryless channels is derived. Such a representation leads to simple upper bounds on the bit error probabilities for code components expressed via 2n code spectra, where n is the code length, and to a "bounded distance decoding" algorithm. The performance of the algorithm is illustrated for the (23,12) Golay code and a particular channel having non-binary output alphabet where it attains a sub-optimum performance.
An efficient algorithm for calculating the ith bit error probability of a binary linear code over the binary symmetric channel (BSC) is presented. It is proved that the exact ith bit error probability of maximum-likelihood (ML) decoding, bounded distance decoding, and symbol-wise maximum a posteriori probability (MAP) decoding can be obtained with time complexity O(n2/sup n-k/), where n and k denote the length and the dimension of the target code. The proposed methods are applicable to any binary linear code with redundancy up to nearly 25-30 bits with a typical personal computer.