Construction of m-Repeated Burst Error Detecting and Correcting Non-binary Linear Codes

Error correcting codes are required to ensure reliable communication of digitally encoded information. One of the areas of practical importance in which a parallel growth of the subject error correcting codes took place is that of burst error detecting and correcting codes. The nature of burst errors differs from channel to channel depending upon the behavior of channels or the kind of errors which occur during the process of transmission. The rate of transmission is efficient if the number of parity-check digits are as minimum as possible. It is usually not possible to give the exact number of parity-check digits required for a given code. However, bounds can be obtained over the number of parity-check digits. An upper bound for a linear code capable of detecting/ correcting burst errors or its variants is many a times established by the technique used to establish Varsharmov-Gilbert-Sacks bound by constructing a parity-check matrix for the requisite code. This technique not only ensures the existence of such a code but also gives a method for constructing such a code. The synthesis method using this technique is cumbersome and to the best of our knowledge, there is no systematic way to construct a parity-check matrix for a burst error correcting non-binary linear code. Extending the algorithm for binary linear codes given by the authors to non-binary codes, the paper proposes a new algorithm for constructing a parity-check matrix for any linear code over GF(q) capable of detecting and correcting a new kind of burst error called `m-repeated burst error of length b or less\u27. Codes based on the proposed algorithm have been illustrated

On repeated low-density burst error detecting linear codes

The paper presents lower and upper bounds on the number of parity-check digits required for a linear code that is capable of detecting repeated low-density burst errors of length $b$ (fixed) with weight w or less $(w leq b)$. A bound for codes which can correct and simultaneously detect such burst errors has also been derived. An illustration has been provided for the code detecting 2-repeated burst errors of length 3 (fixed) with weight 2 or less over GF(2)

Codes on m-repeated solid burst errors

In coding theory, several kinds of errors due to the different behaviours of communication channels have been considered and accordingly error detecting and error correcting codes have been constructed. In general communication due to the long messages, the strings of same type of error may repeat in a vector itself. The concept of repeated bursts is introduced by Beraradi, Dass and Verma [4] which has opened a new area of study. They defined 2-repeated bursts and obtained results for detection and correction of such type of errors. The study was further extended to m-repeated bursts [3]. Solid burst errors are common in many communications. This paper considers a new similar kind of error which will be termed as ‘m-repeated solid burst error of length b’. A lower bound on the number of parity checks required for the existence of codes that detect such errors is obtained. Further, codes capable of detecting and simultaneously correcting such errors have also been dealt with.Publisher's Versio

On 2-Repeated Burst Codes

There are several kinds of burst errors for which error detecting and error correcting codes have been constructed. In this paper, we consider a new kind of burst error which will be termed as ‘2-repeated burst error of length b(fixed)’. Linear codes capable of detecting such errors have been studied. Further, codes capable of detecting and simultaneously correcting such errors have also been dealt with. The paper obtains lower and upper bounds on the number of parity-check digits required for such codes. An example of such a code has also been provided

Blockwise Repeated Burst Error Correcting Linear Codes

This paper presents a lower and an upper bound on the number of parity check digits required for a linear code that corrects a single sub-block containing errors which are in the form of 2-repeated bursts of length b or less. An illustration of such kind of codes has been provided. Further, the codes that correct m-repeated bursts of length b or less have also been studied

Codes on s-periodic errors

In this paper, we study linear codes capable of detecting and correcting s-periodic errors. Lower and upper bounds on the number of parity check digits required for codes detecting such errors are obtained. Another bound on codes correcting such errors is also obtained. An example of a code detecting such errors is provided

Error Locating Codes Dealing with Repeated Low-Density Burst Errors

This paper presents a study of linear codes which are capable to detect and locate errors which are repeated low-density bursts of length b(fixed) with weight w or less. An illustration for such a kind of code has also been provided

Location of burst and repeated burst error in single and adjacent sub-blocks

The paper gives necessary and sufficient conditions for the existence of linear codes capable of identifying burst/repeated burst errors whether it is confined to one sub-block or spread over two adjacent sub-blocks. Examples of such codes are also provided. We also provide two methods one using tensor product and other using cyclic code to construct such codes. Finally, comparisons on the number of check digits of such codes with the corresponding error detecting and correcting codes are also provided.Publisher's Versio