Burst error-correcting code
In coding theory, burst error-correcting codes employ methods of correcting burst errors, which are errors that occur in many consecutive bits rather than occurring in bits independently of each other.
Examples of burst errors can be found extensively in storage mediums.
These errors may be due to physical damage such as scratch on a disc or a stroke of lightning in case of wireless channels.
To remedy the issues that arise by the ambiguity of burst descriptions with the theorem below, however before doing so we need a definition first.
where the individual symbols of a word correspond to the different coefficients of the polynomial.
The above proof suggests a simple algorithm for burst error detection/correction in cyclic codes: given a transmitted word (i.e. a polynomial of degree
This contradicts the Distinct Cosets Theorem, therefore no nonzero burst of length
distinct symbols, otherwise, the difference of two such polynomials would be a codeword that is a sum of two bursts of length
is called the redundancy of the code and in an alternative formulation for the Abramson's bounds is
, we mean that all errors that a received codeword possess lie within a fixed span of
The reason is simple: we know that each coset has a unique syndrome decoding associated with it, and if all bursts of different lengths occur in different cosets, then all have unique syndromes, facilitating error correction.
, representing the burst error correction capability of our code, and we need to satisfy the property that
Certain families of codes, such as Reed–Solomon, operate on alphabet sizes larger than binary.
This property awards such codes powerful burst error correction capabilities.
The reason such codes are powerful for burst error correction is that each symbol is represented by
The basic idea behind the use of interleaved codes is to jumble symbols at the transmitter.
Thus, the main function performed by the interleaver at transmitter is to alter the input symbol sequence.
This is obvious from the fact that we are reading the output column wise and the number of rows is
): It is found by taking ratio of burst length where decoder may fail to the interleaver memory.
Thus, these factors give rise to two drawbacks, one is the latency and other is the storage (fairly large amount of memory).
): It is found by taking the ratio of burst length where decoder may fail to the interleaver memory.
Thus, we need to store maximum of around half message at receiver in order to read first row.
Without error correcting codes, digital audio would not be technically feasible.
This makes the RS codes particularly suitable for correcting burst errors.
[3] Current compact disc digital audio system was developed by N. V. Philips of The Netherlands and Sony Corporation of Japan (agreement signed in 1979).
A compact disc comprises a 120 mm aluminized disc coated with a clear plastic coating, with spiral track, approximately 5 km in length, which is optically scanned by a laser of wavelength ~0.8 μm, at a constant speed of ~1.25 m/s.
), disc handling (scratches – generally thin, radial and orthogonal to direction of recording) and variations in play-back mechanism.
CIRC (Cross-Interleaved Reed–Solomon code) is the basis for error detection and correction in the CD process.
It corrects error bursts up to 3,500 bits in sequence (2.4 mm in length as seen on CD surface) and compensates for error bursts up to 12,000 bits (8.5 mm) that may be caused by minor scratches.
It is up to individual designers of CD systems to decide on decoding methods and optimize their product performance.
