Showing papers on "List decoding published in 1973"
TL;DR: A new lower bound on the probability of decoding error for the case of rates above capacity is presented, which forms a natural companion to Gallager's random coding bound for rates below capacity.
Abstract: A new lower bound on the probability of decoding error for the case of rates above capacity is presented. It forms a natural companion to Gallager's random coding bound for rates below capacity. The strong converse to the coding theorem follows immediately from the proposed lower bound.
241 citations
TL;DR: The result shows that the weakness of the random coding bound at rates below the second critical rate is due not to upperbounding the ensemble average, but rather to the fact that the best codes are much better than the average at low rates.
Abstract: The random coding bound of information theory provides a well-known upper bound to the probability of decoding error for the best code of a given rate and block length. The bound is constructed by upper-bounding the average error probability over an ensemble of codes. The bound is known to give the correct exponential dependence of error probability on block length for transmission rates above the critical rate, but it gives an incorrect exponential dependence at rates below a second lower critical rate. Here we derive an asymptotic expression for the average error probability over the ensemble of codes used in the random coding bound. The result shows that the weakness of the random coding bound at rates below the second critical rate is due not to upperbounding the ensemble average, but rather to the fact that the best codes are much better than the average at low rates.
91 citations
TL;DR: The concept of combining the design of the error-correcting-coding approach and the modulation format is treated in general terms and its effectiveness is demonstrated by a specific implementation referred to as Codem I.
Abstract: The concept of combining the design of the error-correcting-coding approach and the modulation format is treated in general terms and its effectiveness is demonstrated by a specific implementation referred to as Codem I. A new decoding algorithm is presented, which has the interesting property that the only channel measurement information utilized is the relative reliability of each received digit. The effectiveness of this decoding technique is demonstrated by computer simulation over three different channel models for dispersive channels. Comparisons of the performance of Codem I with the more conventional 16-tone (four-phase DPSK) HF modem are obtained by actual field results as well as by computer simulations. Improvement in error probability in the region of two orders of magnitude is demonstrated when both systems are operating under similar channel conditions and at equal data rates. A further improvement is demonstrated when channel measurement information is used to reject a small percentage (typically less than 3 percent) of codewords that are considered unreliable.
90 citations
01 Jul 1973
TL;DR: A variable-word-length minimum-redundant code is described that has the advantages of both the Huffman and Shannon-Fano codes in that it reduces transmission time, storage space, translation table space, and encoding and decoding times.
Abstract: A variable-word-length minimum-redundant code is described. It has the advantages of both the Huffman and Shannon-Fano codes in that it reduces transmission time, storage space, translation table space, and encoding and decoding times.
67 citations
TL;DR: The use of majority logic decoding as a pseudonoise code acquisition technique is considered and it is shown that the probability of acquiring an 8191 code in one attempt can be made nearly one at -10-dB SNR.
Abstract: This paper considers the use of majority logic decoding as a pseudonoise code acquisition technique A bound on the probability of code acquisition is derived and it is shown that the probability of acquiring an 8191 code in one attempt can be made nearly one at -10-dB SNR
51 citations
TL;DR: The method of sequential encoding and decoding is generalized to the case of a source with redundancy and a computational entropy analogous to the computational cutoff rate of the channel is introduced.
Abstract: The method of sequential encoding and decoding is generalized to the case of a source with redundancy. A computational entropy of the source analogous to the computational cutoff rate of the channel is introduced. A range of transmission rates is found for which the average number of decoding computations is finite.
20 citations
TL;DR: Certain classes of low-rate binary codes that have simple decoding algorithms can be used as underlying codes in the construction of high-rate easily decodable i -compressed codes, which have higher rates than binary codes of comparable length and number of correctable errors.
Abstract: In this paper we present a new error-control technique intended for use in 2^l -level data-transmission systems that employ Gray coding to transform a binary source sequence into the 2^l -ary transmitted sequence. The codes, which we call i -compressed codes, make use of the structure of binary codes and have the property that for some integer i , 1 \leq i \leq l , transmission errors can be corrected if the erroneously received signals lie less than 2^{i-1} levels from the corresponding correct, or nominal signal levels. The number of such errors that can be corrected is related to the error-correcting capability of the underlying binary code used in the construction. In return for this restriction on the magnitude of correctable errors in the received signal, these codes have higher rates than binary codes of comparable length (in bits) and number of correctable errors. Hence in applications where it can be assumed that the fraction of errors exceeding a certain magnitude is negligible (or at least tolerable), this technique is more efficient than the conventional practice of placing a binary encoder between the data source and modulator and a binary decoder between the demodulator and data sink. Furthermore, although the i -compressed codes are nonbinary, the decoding algorithm is that of the underlying binary code plus a small amount of additional processing; hence it is generally simpler to implement than other nonbinary decoding algorithms. It is also observed that the rate of an i -compressed code is always greater than that of the underlying binary code. Thus certain classes of low-rate binary codes that have simple decoding algorithms can be used as underlying codes in the construction of high-rate easily decodable i -compressed codes. Finally, for the case i = 1 , encoding and decoding becomes exceptionally simple and for this case it is possible to make use of "soft decisions" at the receiver to improve the performance.
17 citations
TL;DR: A general decoding method for cyclic codes is presented which gives promise of substantially reducing the complexity of decoders at the cost of a modest increase in decoding time (or delay).
Abstract: A general decoding method for cyclic codes is presented which gives promise of substantially reducing the complexity of decoders at the cost of a modest increase in decoding time (or delay). Significant reductions in decoder complexity for binary cyclic finite-geometry codes are demonstrated.
12 citations
09 Jul 1973
TL;DR: The properties of linear cyclic codes are considered in detail and new results on construction and analysis of the properties of such codes are presented, as well as basic methods of decoding, using the algebraic properties of codes.
Abstract: : The report is a Russian translation devoted to methods of construction and decoding of cyclic correction codes. The properties of linear cyclic codes are considered in detail and new results on construction and analysis of the properties of such codes are presented. Basic methods of decoding, using the algebraic properties of codes, are outlined as follows: A method based on use of linear multistage filters and a selector; A method based on the properties of symmetry of linear codes; A method of direct and step decoding for Bose-Chaudhuri-Hockquenghem codes. Majority decoding, which has a very simple technical realization, is considered in detail. Corresponding codes and decoding schemes based on the apparatus of finite projective geometries are described. The book is intended for scientific researchers, engineers and students, working in the field of digital data transmission through noisy communications channels, as well as mathematicians who are interested in the use of algebraic methods.
4 citations
TL;DR: A technique is presented for the algebraic decoding of block codes over a, q -ary input, Q -ary output channel ( Q > q ) where Hamming distance, Lee distance, or a burst distance can be assumed.
Abstract: A technique is presented for the algebraic decoding of block codes over a, q -ary input, Q -ary output channel ( Q > q ). It is assumed that an algebraic decoding algorithm is known for a simple channel such as a channel where the input alphabet is identical to the output alphabet. This decoding algorithm is then adapted for use over the actual channel. The technique can be used in conjunction with an arbitrary distance measure between input and output vectors. Thus, Hamming distance, Lee distance, or a burst distance can be assumed. Examples are presented for each of these distances.
01 Jan 1973
TL;DR: In this paper, a finite system of distinct words in the alphabet A, a free semigroup over A, [U] a subsemigroup of ǫ, generated by the set U, and λ the empty word.
Abstract: Let U be a finite system of distinct words in the alphabet A, 𝔄 a free semigroup over A, [U] a subsemigroup of 𝔄, generated by the set U, and λ the empty word. By ||X|| we shall denote the number of elements of the set X, and by |x| the length of the word x.
TL;DR: It is shown that a majority-logic decoding algorithm proposed by Lin and Weldon for the product of an L -step and a one-step orthogonalizable code is incomplete when L is greater than unity, and an improvement is presented to overcome this disadvantage.
Abstract: It is shown that a majority-logic decoding algorithm proposed by Lin and Weldon for the product of an L -step and a one-step orthogonalizable code is incomplete when L is greater than unity. An improvement is presented to overcome this disadvantage in the binary case.