By Richard E. Blahut

Error-correcting codes play a primary position in glossy communications and data-storage structures. This quantity presents an available creation to the elemental parts of algebraic codes and discusses their use in numerous purposes. the writer describes quite a number vital coding options, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. during the publication, mathematical conception is illustrated through connection with many useful 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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**Example text**

Ai j ]. . . an1 an2 . . anm In most applications, the ring R is actually a ﬁeld, and we shall usually restrict our attention to this case. We are mostly concerned with matrices over a ﬁnite ﬁeld G F(q). The set of elements aii , for which the column number and row number are equal, is called the main diagonal. If all elements not on the main diagonal are zero, the matrix is called a diagonal matrix. If n equals m, the matrix is called a square matrix of size n. An n by n matrix with the ﬁeld element one in every entry of the main diagonal, and the ﬁeld element zero in every other matrix entry, is called an n by n identity matrix.

Multiplication of any row by a nonzero ﬁeld element. 3. Replacement of any row by the sum of that row and a multiple of any other row. 3 If a matrix is deﬁned over a commutative ring with identity, the matrix inverse exists if and only if det(A) is a unit of the ring. 6 Linear algebra Each elementary row operation is inverted by an elementary row operation of the same kind. Each elementary row operation on an n by m matrix A can be effected by multiplying the matrix A from the left by an appropriate n by n matrix F, called an elementary matrix.

Give two doubleerror patterns that cannot be distinguished from one another when using this code and so cannot be corrected. d. What is the minimum distance of the code? 7 Show that Hamming distance has the following three properties: (i) d(x, y) ≥ 0 with equality if and only if x = y; (ii) d(x, y) = d(y, x). 19 Problems (iii) Triangle inequality d(x, y) ≤ d(x, z) + d(y, z). A distance function with these three properties is called a metric. 8 a. Show that a code C is capable of detecting any pattern of d or fewer errors if and only if the minimum distance of the code C is greater than d.