Repeated Burst Error Correcting Linear Codes

Abstract

Many kinds of errors in coding theory have been dealt with for which codes have been constructed to combat such errors. Though there is a long history concerning the growth of the subject and many of the codes developed have found applications in numerous areas of practical interest, one of the areas of practical importance in which a parallel growth of the subject took place is that of burst error detecting and correcting codes. The nature of burst errors differ from channel to channel depending upon the behaviour of channels or the kind of errors which occur during the process of data transmission. In very busy communication channels, errors repeat themselves more frequently. In view of this, it is desirable to consider repeated burst errors. The paper presents lower and upper bounds on the number of parity-check digits required for a linear code correcting errors in the form of repeated bursts. An upper bound for a code that detects m-repeated bursts has also been derived. Illustrations of several codes that correct 2-repeated bursts of different lengths have also been given.

On 2-Repeated Solid Burst Errors

“…This is an extension of the idea of open-loop burst given by Fire [4]. Later on Dass and Verma [2] defined m-repeated bursts and obtained results regarding the number of parity-check digits required for codes detecting such errors. Some results have been obtained on weights of such errors by Sharma and Rohtagi [11].…”

Bounds on linear codes capable of detecting, locating and correcting of repeated burst errors prevailing in multiple sub-blocks

“…Let us suppose that all columns of f − 1 sub-blocks and first j − 1 columns of the f th sub-block are chosen suitably and added to H. Now, we put down the condition to add the j th column h j such that the column h j should not be a linear combination of immediately preceding b − 1 columns of the f th sub-block, together with m − 1 sets of b or fewer successive columns out of the first j − b columns of the f th sub-block, along with any m sets of b or fewer successive columns within a sub-block chosen from any l − 1 or less sub-blocks out of the remaining f − 1 sub-blocks. In other words, we can write this condition as follows: In the expression (2.3), the total choices of a i 's is q b−1 and the choices of c i 's is equal to the number of m − 1 repeated bursts of length b or less in a (j − b)-tuple which is (see [5])…”

“…Blockwise correction of errors is very useful specially in the study of convolutional codes [14,10], where the codewords consist of some smaller sub-blocks. Repeated burst error was introduced by Berardi, Dass and Verma in [1] and its general form was introduced by Dass and Verma in [5]. Repeated burst error affecting only single sub-block is studied by Dass and Madan in [7,8] where they obtain bounds for linear code correcting and locating such errors.…”

Power-Aware Wireless Communication System Design for Body Area Networks

“…BCH code is cyclic codes and underlies in the class of linear block codes, also contains the message bit and the parity bit [20]. It can be used to correct the errors which are derived from the noise and the channel interference, especially the burst errors [21]. BCH (31,19) code, a short code, and BCH (63,51) code, are employed in the IEEE 802.15.6 standard.…”