# Concatenated kernel codes

22 Jun 2020-Discrete Mathematics, Algorithms and Applications (World Scientific Publishing Company)-Vol. 12, Iss: 03, pp 2050044

TL;DR: An example of concatenated kernel code and its trellis is constructed to demonstrate the importance of defined code andIts computation.

Abstract: Concatenated codes introduced by Forney in 1966 received wide attention due to their extensive usage in space missions. Thereafter, many concatenated codes were constructed on the similar lines and...

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31 Jul 1996

TL;DR: In this article, the authors investigated the use of modulation block codes as the inner code of a concatenated coding system in order to improve the overall space link communications performance and identified and analyzed candidate codes that will complement the performance of the overall coding system which uses the interleaved RS (255,223) code as the outer code.

Abstract: This report describes the progress made towards the completion of a specific task on error-correcting coding. The proposed research consisted of investigating the use of modulation block codes as the inner code of a concatenated coding system in order to improve the overall space link communications performance. The study proposed to identify and analyze candidate codes that will complement the performance of the overall coding system which uses the interleaved RS (255,223) code as the outer code.

179 citations

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TL;DR: In this paper , an error correction code which is capable of correcting indel errors and maintaining the stability of DNA strings was constructed by using Varshamov-Tenengolts algorithm.

Abstract: A procedure for storage and retrieval of Digital information in DNA strings is discussed by constructing an error correction code which is capable of correcting indel errors and maintaining the stability of DNA strings. For correction of indel errors, Varshamov-Tenengolts algorithm is used. To maintain the stability of DNA code, Reverse-complementary property, and GC-content constraints are implemented by appropriate choices of group homomorphisms. Code for any desired length(n) can be created using the construction methods presented in this paper. The code's reverse complement distance is calculated using the value of n.

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TL;DR: The author was led to the study given in this paper from a consideration of large scale computing machines in which a large number of operations must be performed without a single error in the end result.

Abstract: The author was led to the study given in this paper from a consideration of large scale computing machines in which a large number of operations must be performed without a single error in the end result. This problem of “doing things right” on a large scale is not essentially new; in a telephone central office, for example, a very large number of operations are performed while the errors leading to wrong numbers are kept well under control, though they have not been completely eliminated. This has been achieved, in part, through the use of self-checking circuits. The occasional failure that escapes routine checking is still detected by the customer and will, if it persists, result in customer complaint, while if it is transient it will produce only occasional wrong numbers. At the same time the rest of the central office functions satisfactorily. In a digital computer, on the other hand, a single failure usually means the complete failure, in the sense that if it is detected no more computing can be done until the failure is located and corrected, while if it escapes detection then it invalidates all subsequent operations of the machine. Put in other words, in a telephone central office there are a number of parallel paths which are more or less independent of each other; in a digital machine there is usually a single long path which passes through the same piece of equipment many, many times before the answer is obtained.

5,408 citations

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01 Apr 1990TL;DR: This work presents general techniques for constructing simple to program self-testing/correcting pairs for a variety of numerical functions, including integer multiplication, modular multiplication, matrix multiplication, inverting matrices, computing the determinant of a matrix, Computing the rank of a Matrix, integer division, modular exponentiation, and polynomial multiplication.

Abstract: Suppose someone gives us an extremely cast program P that we can call as a black box to compute a function f. Should we trust that P works correctly? A self-testing/correcting pair for f allows us to: (1) estimate the probability that P(x) ¬= f(x) when x is randomly chosen; (2) on any input x, compute f(x) correctly as long as P is not too faulty on average. Furthermore, both (1) and (2) take time only slightly more than the original running time of P. We present general techniques for constructing simple to program self-testing/correcting pairs for a variety of numerical functions, including integer multiplication, modular multiplication, matrix multiplication, inverting matrices, computing the determinant of a matrix, computing the rank of a matrix, integer division, modular exponentiation, and polynomial multiplication

913 citations

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676 citations

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TL;DR: It is shown that minimal proper trellises exists for all block codes and bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes.

Abstract: Basic concepts in the study of trellises of block codes are defined. It is shown that minimal proper trellises exists for all block codes. Bounds on the size of such trellises are established. These bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes. >

238 citations

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27 Jun 1994TL;DR: The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NP-complete and the number of distinct minimal linear block code trellises is a Stirling number of the second kind.

Abstract: The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NP-complete. Examples are provided that show that (1) the minimal trellis for a nonlinear code may not be observable, i.e. some codewords may be represented by more than one path through the trellis and (2) minimizing the vertex count at one time index may be incompatible with minimizing the vertex count at another time index. A trellis produce is defined and used to construct trellises for sum codes. Minimal trellises for linear codes are obtained by forming the product of elementary trellises corresponding to the one-dimensional subcodes generated by atomic codewords. The structure of the resulting trellis is determined solely by the spans of the atomic codewords. A correspondence between minimal linear block code trellises and configurations of nonattacking rooks on a triangular chess board is established and used to show that the number of distinct minimal linear block code trellises is a Stirling number of the second kind. Various bounds on trellis size are reinterpreted in this context.

200 citations