By Paul B. Bailey

ISBN-10: 0120733501

ISBN-13: 9780120733507

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**Extra resources for Nonlinear 2 Point Boundary Value Problems **

**Sample text**

Clearly (Ca-)a- = C. Therefore if H is any parity check matrix of C, the code C can also be defined as C - - { x E I F n : H x T - 0). For every x E IF '~, we call the vector H x T C IFn-k the syndrome of x. Hence the code consists of the vectors with syndrome equal to 0. , the minimum weight of a nonzero vector x E IFq such that H x T = 0. In particular, in the binary case we have the following theorem. 8 Let C be a binary [n, k] code with parity check matriz H. The m i n i m u m distance of C is the smallest positive integer d such that the sum of some d columns of H is O.

However, assume that we have an efficient algorithm for correcting up to e' _< e errors. The algorithm gets as an input any binary vector of length n, and outputs the unique codeword within Hamming distance e' or tells us that no such codeword exists. We wish to use the algorithm for correcting t < e' errors and f erasures, where 2t + f < d. , replaced by 9indicating that we do not know if the symbol is 0 or 1, and in at most t places 0 is replaced by 1 or 1 is replaced by 0. For notational convenience, let us assume that the erasures occur among the first f coordinates and that v is the binary vector formed by the last n - f coordinates in the received vector.

If d or R are not needed, we call (7 an (n, K)q, (n, K, d)q or (n, K)qR code. The subscript q is usually omitted if q-2. Chapter 2. Basic facts 18 We denote Kq(n, R) - m i n { K (n, K)qR code) 9there is an for 0 < R < n. An (n, K)qR code with cardinality K - Kq(n, R) is caned optimal. An (n,K,d)qR code C C Qn is called mazimalifR < d - 1, or equivalently, if for every x E Q'~ the code C u { x ) has m i n i m u m distance strictly less than d. We also denote tq(n, K ) - min{R and Aq(n, d) - m a x { K 9there is an (n, K)qR code} 9there is an (n, K, d)q code}.

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