On a new code,[2n-1, n, 2n-1] (original) (raw)

Linear Codes with Non-Uniform Error Correction Capability

Designs, Codes and Cryptography, 1997

This paper introduces a class of linear codes which are non-uniform error correcting, i.e. they have the capability of correcting different errors in different codewords. A technique for specifying error characteristics in terms of algebraic inequalities, rather than the traditional spheres of radius e, is used. A construction is given for deriving these codes from known linear block codes. This is accomplished by a new method called parity sectioned reduction. In this method, the parity check matrix of a uniform error correcting linear code is reduced by dropping some rows and columns and the error range inequalities are modified.

Error-Correction Capability of Binary Linear Codes

IEEE Transactions on Information Theory, 2005

The monotone structure of correctable and uncorrectable errors given by the complete decoding for a binary linear code is investigated. New bounds on the error-correction capability of linear codes beyond half the minimum distance are presented, both for the best codes and for arbitrary codes under some restrictions on their parameters. It is proved that some known codes of low rate are as good as the best codes in an asymptotic sense.

SOME NEW RESULTS ON BINARY LINEAR BLOCK CODES

Certain properties of the parity-check matrix H of (n, k) linear codes are used to establish a computerised search procedure for new binary linear codes. Of the new error-correcting codes found by this procedure, two codes were capable of correcting up to two errors, three codes up to three errors, four codes up to four errors and one code up to five errors. Two meet the lower bound given by Helgert and Stinaff, and seven codes exceed it. In addition, one meets the upper bound. Of the even-Hamming-distance versions of these codes, eight meet the upper bound, and the remaining two exceed the lower bound.

Minimum-distance bounds for binary linear codes

IEEE Transactions on Information Theory, 1973

This paper presents a table of upper and lower bounds on rl,,,(rr,k), the maximum minimum distance over all binary, linear (n,k) error-correcting codes. The table is obtained by combining the best of the existing bounds on d,,,(n,k) with the mini&n distances of known codes and a variety of code-construction techniques.

Binary and ternary linear codes which are good and proper for error correction

All binary cyclic codes of lengths up to 31 and ternary cyclic and negacyclic codes of lengths up to 20 have been tested and those of them which satisfy the sufficient conditions to be good and proper for error correction have been determined. The same way some binary distance optimal linear codes of lengths up to 33 have been tested. Tables with the results have been prepared.

Optimal linear codes with a local-error-correction property

2012 IEEE International Symposium on Information Theory Proceedings, 2012

Motivated by applications to distributed storage, Gopalan et al recently introduced the interesting notion of information-symbol locality in a linear code. By this it is meant that each message symbol appears in a parity-check equation associated with small Hamming weight, thereby enabling recovery of the message symbol by examining a small number of other code symbols. This notion is expanded to the case when all code symbols, not just the message symbols, are covered by such "local" parity. In this paper, we extend the results of Gopalan et. al. so as to permit recovery of an erased code symbol even in the presence of errors in local parity symbols. We present tight bounds on the minimum distance of such codes and exhibit codes that are optimal with respect to the local error-correction property. As a corollary, we obtain an upper bound on the minimum distance of a concatenated code.

A Construction of Binary Punctured Linear Codes and A Supporting Method for Best Code Search

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 2021

Reduction of redundancy and improvement of errorcorrecting capability are essential research themes in the coding theory. The best known codes constructed in various ways are recorded in a database maintained by Markus Grassl. In this paper, we propose an algorithm to construct the best code using punctured codes and a supporting method for constructing the best codes. First, we define a new evaluation function to determine deletion bits and propose an algorithm for constructing punctured linear codes. 27 new best codes were constructed in the proposed algorithm, and 112 new best codes were constructed by further modifying those best codes. Secondly, we evaluate the possibility of increasing the minimum distance based on the relationship between code length, information length, and minimum distance. We narrowed down the target (n, k) code to try the best code search based on the evaluation and found 28 new best codes. We also propose a method to rapidly derive the minimum weight of the modified cyclic codes. A cyclic code loses its cyclic structure when it is modified, so we extend the k-sparse algorithm to use it for modified cyclic codes as well. The extended k-sparse algorithm is used to verify our newly constructed best code.

Construction of Error Control Run Length Limited Codes Exploiting Some Parity Matrix Properties

Journal of Electrical Engineering, 2015

Error control codes (ECC) as well as translation codes (TC) are used today in many different systems such as computer storages, communications systems and consumer electronic devices. ECC introduce redundancy into the encoded digital sequence in order to decrease the number of errors at output of its decoder [1]. TC introduce redundancy, in order to translate any digital sequence at the input of TC encoder to such output sequence, which fulfills constrains deduced from practical requirements. It is possible to construct codes, which have both of these properties, so called Transcontrol codes or their subclass error control run length limited (ECRLL) codes. In this manuscript a new approach to construction of EC-RLL codes is presented. The new construction is based on some parity check matrix properties of a linear binary block code from which the new EC-RLL code is obtained.