By Richard E. Blahut
Error-correcting codes play a primary position in sleek communications and data-storage structures. This quantity offers an available creation to the fundamental components of algebraic codes and discusses their use in quite a few purposes. the writer describes a variety of very important coding concepts, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. through the publication, mathematical conception is illustrated by means of connection with many sensible examples. The booklet is written for graduate scholars of electric and laptop engineering and practising engineers whose paintings consists of communications or sign processing.
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The polynomial D(x) has at most µ − 1 zeros. The coefﬁcient dµ−1 is itself the determinant of a Vandermonde matrix, and by the induction hypothesis, is nonzero. If for any i, 2 ≤ i ≤ µ, we set x = X i , then two rows of the matrix are equal, and D(X i ) = 0. Thus for each i = 1, X i is a zero of D(x), and because they are all distinct and there are µ − 1 of them, the polynomial can be easily factored: µ D(x) = dµ−1 (x − X i ) . i=2 Therefore the determinant of the original Vandermonde matrix is µ D(X 1 ) = dµ−1 (X 1 − X i ) .
Find a set of sixteen binary words of length 7 such that each word is at least a distance of 3 from every other word. 2 a. 5 inches and length 11 inches. Prove that it is not possible to cut out more than 119 circles of 1-inch diameter. b. Prove that it is not possible to ﬁnd 32 binary words, each of length 8 bits, such that every word differs from every other word in at least three places. 3 A single-error-correcting Hamming code has 2m − 1 bits of which m bits are check bits. a. Write (n, k) for the ﬁrst ﬁve nontrivial Hamming codes (starting at m = 3).
J = 1, . . , m. This matrix product is denoted as C = AB. With this deﬁnition of matrix multiplication, and the earlier deﬁnition of matrix addition, the set of n by n square matrices over any ﬁeld F forms a ring, as can be easily veriﬁed. It is a noncommutative ring, but it does have an identity, namely the n by n identity matrix. A matrix can be partitioned as follows: A= A11 A21 A12 A22 where A11 , A12 , A21 , and A22 are smaller matrices whose dimensions add up to the dimensions of A in the obvious way.
Algebraic codes for data transmission by Richard E. Blahut