Binary Burst Error Correcting Cyclic Codes Designed with the Circulant Parity Check Matrix

Byte correcting perfect codes are developed to correct burst errors within bytes. If a code is byte correcting code and we say that the code is [Formula: see text]-burst correcting, meaning that it corrects a single burst of length [Formula: see text] or less within a byte. A byte correcting code is such that if [Formula: see text]; [Formula: see text] denote the set of syndromes obtained from the [Formula: see text]th-byte of the parity check matrix [Formula: see text] and [Formula: see text]; [Formula: see text] denote the set of syndromes obtained from the [Formula: see text]th-byte of the parity check matrix [Formula: see text] then [Formula: see text]. In an [Formula: see text] code, if there are [Formula: see text] bytes of size [Formula: see text], then [Formula: see text]. Byte correcting codes are preferred where information stored in all bytes are equally important. But there are cases where some parts of the message are more important than other parts of the message, for example, if we have to transmit a message on a border location “ Shift Battalion (Bn) from Location A to Location B” then, we will focus more on the information shift, location [Formula: see text] and location [Formula: see text] i.e., bytes containing these information will be more important than others. In this situation, it is needed that during transmission these bytes should have no possibility of error. In other words, these bytes should be protected absolutely against any error. Keeping this in mind, we study burst error correcting capabilities of byte oriented codes in terms of byte protection level of each byte. If there is a byte error pattern of length [Formula: see text] in the transmission then all those bytes of the received pattern will be decoded correctly whose burst protection level is [Formula: see text] or more even though the code word may be decoded wrongly. Taking the code length [Formula: see text] to be divided into [Formula: see text] bytes with burst protection level of the [Formula: see text]th-byte as [Formula: see text]; [Formula: see text]; [Formula: see text], we construct linear codes that we call byte protecting burst (BPB) codes and investigate their byte protecting capabilities in this paper.

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Codes correcting and simultaneously detecting solid burst errors

Computer Engineering and Applications Journal ◽

10.18495/comengapp.v2i1.15 ◽

2013 ◽

Vol 2 (1) ◽

pp. 143-150

Author(s):

P.K. Das

Keyword(s):

Coding Theory ◽

Linear Codes ◽

Communication Channels ◽

Upper Bounds ◽

Parity Check ◽

Lower And Upper Bounds ◽

Parity Check Matrix ◽

Burst Error ◽

Check Matrix ◽

Error Detecting

Detecting and correcting errors is one of the main tasks in coding theory. The bounds are important in terms of error-detecting and -correcting capabilities of the codes. Solid Burst error is common in several communication channels. This paper obtains lower and upper bounds on the number of parity-check digits required for linear codes capable of correcting any solid burst error of length b or less and simultaneously detecting any solid burst error of length s(>b) or less. Illustration of such a code is also provided.Keywords: Parity check matrix, Syndromes, Solid burst errors, Standard arrayDOI:Â 10.18495/comengapp.21.143150Â Â

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CONSTRUCTION OF m-REPEATED BURST ERROR CORRECTING BINARY LINEAR CODE

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830912500437 ◽

2012 ◽

Vol 04 (03) ◽

pp. 1250043 ◽

Cited By ~ 1

Author(s):

B. K. DASS ◽

RASHMI VERMA

Keyword(s):

Linear Code ◽

Error Correcting Code ◽

Binary Codes ◽

Parity Check ◽

Parity Check Matrix ◽

Simple Method ◽

Burst Error ◽

Binary Linear Code ◽

Check Matrix ◽

Standing Problem

The purpose of this paper is to propose a simple method of constructing a parity-check matrix for any binary linear code capable of correcting a new kind of burst error called 'm-repeated burst error of length b or less' recently introduced by the authors. Some binary codes based on the proposed technique have been illustrated. This technique for m = 1 helps in resolving a long standing problem of devising a systematic algorithm for the construction of a burst error correcting code.

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