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AN INVESTIGATION OF METHODS OF DETECTING AND CORRECTING ERRORS IN DATA TRANSMISSION SYSTEMS.

W. W. Peterson, M. J. Moss, J. B. O'Neal, Edward J Weldon

Published Feb 1, 1964• 1 citation
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Abstract

A summary of some computations made on BoseChaudhuri codes is given. It is shown that there are infinitely many codes for which Bose's bound on minimum distance gives the true value. Also discussed are two theorems relating to the relationship of codewords and their weights. A few weak results on a bit-loss channel, and abstracts of scientific reports on a quantum channel and on binary communication with feedback are included. (Author)

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A summary of some computations made on BoseChaudhuri codes is given. It is shown that there are infinitely many codes for which Bose's bound on minimum distance gives the true value. Also discussed are two theorems relating to the relationship of codewords and their weights. A few weak results on a bit-loss channel, and abstracts of scientific reports on a quantum channel and on binary communication with feedback are included. (Author)
There are some bounds that are placed on the size of error correcting codes. Some of the well-known bounds are discussed in the following sections. 6.1 Definitions Let n be fixed, and let M denote the size of a code C (i.e., the number of codewords in C), and d the minimum distance between the codewords. For example, (i) for the code C = {00, 01, 10, 11} : n = 2,M = 4; d = 1; (ii) for the code C = {000, 011, 101, 110} : n = 3,M = 4, d = 2; and (iii) for the code C = {00000, 00111, 11111} : n = 5,M = 3, d = 2. Then for a q-ary (n,M, d) code, the transmission rate (also known as the information rate) is defined by R(C) = log q M n , and the relative minimum distance by δ(C) = (d − 1)/n. Note that this distance is also defined by d/n, but neater formulas are obtained by defining it as (d − 1)/n. For the binary repetition code C = {00 . . .0︸ ︷︷ ︸ 11 . . .1︸ ︷︷ ︸ n }, which is an (n, 2, n) code, we find that R(C) = log 2 2 n = n → 0 and δ(C) = n− 1 n → 0 as n→∞.
Classical Bose-Chaudhuri-Hocquenghem (BCH) codes C that contain their dual codes can be used to construct quantum stabilizer codes this chapter studies the properties of such codes. It had been shown that a BCH code of length n which contains its dual code satisfies the bound on weight of any non-zero codeword in C and converse is also true. One impressive difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. To address this difficulty, many good quantum errorcorrecting codes have been derived as binary stabilizer codes. We were able to shed more light on the structure of dual containing BCH codes. These results make it possible to determine the parameters of quantum BCH codes in terms of weight of non-zero dual codeword.
Polar codes are usually constructed by ranking synthetic bit-channels according to reliability, which guarantees capacity-achieving behavior but can yield poor low-weight spectra at short and moderate lengths. Recent algebraic results express the contribution of individual bit-channels to the multiplicities of minimum and near-minimum weight codewords in closed form. In this work we combine these insights into a mixed (reliability--weight) bit-channel ordering. We define a per-bit cost whose distance term is derived from orbit enumeration of minimum-weight codewords and scaled by a Bhattacharyya-type factor, and show that the resulting mixed construction minimises a truncated SC/ML union-bound surrogate within a class of decreasing monomial codes. We relate the mixed metric to error events in SCL decoding via a pruning/ML decomposition, and prove that mixed designs act as local perturbations of reliability-based constructions whose asymptotic impact vanishes as code-length approaches infinity. Numerical results for short and moderate lengths on BPSK-AWGN, implemented via Gaussian approximation and closed-form weight contributions, illustrate the trade-off between pure reliability-based and mixed constructions in terms of minimum distance, multiplicity, and union-bound approximations. All proofs are deferred to the appendices.
A technique is described for finding simply computable numerical-valued functions of a received binary word whose value indicates where errors in transmission have occurred. Although it seems that a certain condition must usually be fulfilled for such functions to exist, or for our method to constitute an efficient procedure for finding them, there is, on the one hand, a strong tendency for "good" codes to satisfy the condition, while, on the other, it appears to be straightforward to construct codes which are good for a specified channel and also fulfill the condition. An advantage of the resulting decoding procedure is that it corrects and detects all possible errors; more precisely, if a worduis received and the coset\bar{u}to whichubelongs has a unique leadere, the procedure concludes thatu + ewas sent, while ifuhas no unique leader, that fact, along with the weight of\bar{u}(and sometimes a little more) can be indicated. The ideas and techniques are illustrated by the construction of decoding procedures for the perfect (23, 12) three-error-correcting code.