By P. G. Farrell (auth.), G. Longo (eds.)
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Additional info for Algebraic Coding Theory and Applications
The shortest full cyclic cOOe will have =n - c = ip - c. The expcnent to which a polynanial belongs is the least value of p for whidl the polynanial exactly divides ~ + 1, in GF (2). he generator Il'atrix of the cOOe Il'ay be constructed by taking G(x) (in its bincuy form) as the bottan rOIl of [GJ ' xG(x) as the second rOIl, and so on, with x n-c-l G(x) as the top rON. , the code W'Ords are linear canbinations of the rOIlS rGJ will not be in SEE', but by of [G] ) . and canbination, a rOIl interdlange can be Il'ade so (colurm interdlange is not possible, as this destroys the cyclic properties of the cOOe).
Xi can be used for efficient randati error correction provided n/k > t; thus it is best applied only to low rate and/or small t codes. ability of rrore ~rful decoding is used. cyclically ~ft A great deal of the error-correcting In practice it is not necessary to the received sequence, but only the contents . fter shifting. · ve. on ue g m1.. S used . o t errors I Suppose that a cyc li c code can but it cannot be guaranteed that rrore than t-l errors can be trapped (i. , the syndrare weight threshold is t-l).
It is particularly sillple in the case of a cyclic linear code. Thus a:::rcplete decoding can be implemented in bolo stages: (a) syndrare calculation, follONed by (b) determination of the error pattern corresponding to the syndrare. Step (b) can be implemented in a number of ways:- (i) By storing a list of syndro:res and their corresponding error patterns; this, again, is carplex if the number of syndrares exceeds about lOCO. (ii) By systematically searching through all the correctable error patterns, calculating the syndrare for each, until the syndrares agree; the decoding delay of this systematic search algori thIn may be unacceptably large i f the number of correctable patterns is very large.
Algebraic Coding Theory and Applications by P. G. Farrell (auth.), G. Longo (eds.)