Showing papers on "BCH code published in 1972"
TL;DR: The intersection of the binary BCH code of designed distance 2 ^{m-s-1} - 2 ^(s-t-1) - 1 and length 2^m - 1 with the shortened (s + 2) th-order Reed-Muller code of length2^m -- 1 has codewords of weight 2-2-2.
Abstract: For 1 \leq i \leq m - s- 2 and 0 \leq s \leq m -2i , the intersection of the binary BCH code of designed distance 2 ^{m-s-1} - 2 ^{m-s-t-1} - 1 and length 2^m - 1 with the shortened (s + 2) th-order Reed-Muller code of length 2^m -- 1 has codewords of weight 2^{m-s-1} - 2^{m-s-t-1} - 1 .
45 citations
TL;DR: In this correspondence, a decoding algorithm to decode beyond the BCH bound is introduced and gives a complete minimum distance decoding for any cyclic code.
Abstract: In this correspondence, a decoding algorithm to decode beyond the BCH bound is introduced. It gives a complete minimum distance decoding for any cyclic code. A comparison between this decoding algorithm and previously existing ones is also given.
44 citations
TL;DR: Upper and lower bounds on the designed and actual distances of any sequence of extended primitive BCH codes of increasing lengths n and fixed rate R are obtained.
Abstract: In this paper, we obtain upper and lower bounds on the designed and actual distances of any sequence of extended primitive BCH codes of increasing lengths n and fixed rate R . The results of this paper are based on [1, ch. 12], which gives an exact expression for the rates of any sequence of extended primitive BCH codes of increasing length and fixed ratio of distance/length.
34 citations
TL;DR: A survey of results in coding theory obtained since the appearance of Berlekamp's ''Algebraic coding theory'' (1968), concentrating on those which lead to the construction of new codes.
27 citations
TL;DR: A new proof has been found to establish the BCH bound, which relates the minimum distance of a cyclic code to the number of consecutive roots its generator polynomial possesses in a certain field.
Abstract: A new proof has been found to establish the BCH bound, which relates the minimum distance of a cyclic code to the number of consecutive roots its generator polynomial possesses in a certain field. This new proof is recursive in nature and is very simple in structure. It makes no use of the Van der Monde determinant while it unifies the concepts of distance, estimation, and decoding.
22 citations
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TL;DR: A constructive upper bound is obtained on the number of equivalence classes of Srivastava codes, a class of binary SrivASTava codes whose parity-check matrices have row rank less than the maximum, and aclass of binary double-error,correcting codes w!th highly structured parity- check matrices.
Abstract: Srivastava codes, a class of linear noncyclic error-correcting codes, offer performance potentially superior tO that of comparable BCH codes. Their properties are investigated by equivalence classification and subsequent computer evaluation of their weight spectra. In the process a constructive upper bound is obtained on the number of equivalence classes of Srivastava codes, a class of binary Srivastava codes whose parity-check matrices have row rank less than the maximum, and a class of binary double-error,correcting codes w!th highly structured parity-check matrices. A number of shortened binary Srivastava codes with minimum distances superior to those of the best comparable linear codes known are also presented.
19 citations
TL;DR: Two classes of linear error correcting codes are defined which are noncyclic generalizations of the well known BCH and Srivastava codes and lower bounds on the minimum distance of these codes can easily be obtained.
Abstract: We define two classes of linear error correcting codes which are noncyclic generalizations of the well known BCH and Srivastava codes. The corresponding parity check matrices are basically alternating functions of certain elements of GF(qm) and all determinants of maximum order of the parity check matrices are alternants. As a consequence lower bounds on the minimum distance of these codes can easily be obtained.
12 citations
TL;DR: Bose-Chaudhuri-Hocquenghem and Justesen codes are used to pack equa spheres in n –dimensional Euclidean space with density Δ satisfying for all sufficiently large n of the form m 2 m, appearing to be the densest packings yet constructed in high dimensional space.
Abstract: . Bose-Chaudhuri-Hocquenghem and Justesen codes are used to pack equa spheres in n –dimensional Euclidean space with density Δ satisfying for all sufficiently large n of the form m 2 m , where m is a power of 4. These appear to be the densest packings yet constructed in high dimensional space.
12 citations
TL;DR: This correspondence presents a slight rephrasing of the algorithm for binary codes, which results in a very simple circuit for controlling the branching process and performs all necessary tests on the validity of the resulting error-locator polynomial.
Abstract: At each step in Berlekamp's iterative algorithm for BCH codes, the decoder follows one of two possible branches. This correspondence presents a slight rephrasing of the algorithm for binary codes, which results in a very simple circuit for controlling the branching process. This circuit also performs all necessary tests on the validity of the resulting error-locator polynomial.
5 citations
TL;DR: It is shown how binary polynomial residue codes which are equivalent in error-correcting power to shortened Reed-Solomon (R-S) codes can be decoded efficiently with binary operations using the Berlekamp algorithm.
Abstract: It is shown how binary polynomial residue codes which are equivalent in error-correcting power to shortened Reed-Solomon (R-S) codes can be decoded efficiently with binary operations using the Berlekamp algorithm. For R-S codes correcting single error bursts, it is shown how the Chien search can be reduced by reducing the number of points substituted in the error location polynomial. For certain cases, the amount of multiplications needed to evaluate the error location polynomial at a given element of a Galois field can be reduced. This would apply to all BCH codes.
TL;DR: An algorithm is presented for the decoding of triple-error-correcting binary b.c.h that is particularly suitable for parallel implementation and requires no invertors.
Abstract: An algorithm is presented for the decoding of triple-error-correcting binary b.c.h, codes. The algorithm is particularly suitable for parallel implementation and requires no invertors.
TL;DR: In this correspondence the Mattson-Solomon formulation is applied to cyclic codes with multiple sets of consecutive roots of the generator polynomial and better estimates of the minimum distance of manycyclic codes are obtained.
Abstract: In this correspondence the Mattson-Solomon formulation is applied to cyclic codes with multiple sets of consecutive roots of the generator polynomial and better estimates of the minimum distance of many cyclic codes are obtained.
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TL;DR: This paper points out that the algorithm for constructing BCH codes can be applied when the number of levels is a prime or a power of a prime.
Abstract: The use of multi-level p.s.k. arouses interest in non-binary error-correcting codes. This paper points out that the algorithm for constructing BCH codes can be applied when the number of levels is a prime or a power of a prime. Quaternary codes are examined. Possible extensions to higher powers of two and to decimal codes are indicated.
01 Jun 1972
TL;DR: A partitioned 3 state Gilbert model is used to model a burst channel and a method of calculating error sequence probabilities using this model is introduced, and Observations are made about the general type of decoding rule to use to give the lowest probability of decoding error on burst channels when using an interleaving technique.
Abstract: : A brief discussion of basic encoding and decoding on noisy channels is presented to provide a background for the experimental portion of this research. A partitioned 3 state Gilbert model is used to model a burst channel and a method of calculating error sequence probabilities using this model is introduced. Error sequence probability calculations are made using a (7.3) maximal length code and a (15,7) BCH code. Observations are made about the general type of decoding rule to use to give the lowest probability of decoding error on burst channels when using an interleaving technique.