By Richard E. Blahut

Error-correcting codes play a primary position in sleek communications and data-storage platforms. This quantity offers an available creation to the elemental components of algebraic codes and discusses their use in various functions. the writer describes a number very important coding recommendations, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. during the booklet, mathematical idea is illustrated by means of connection with many functional 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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A vector v is in W ⊥ if GvT = 0 because then it is orthogonal to every vector of a basis for W . Let {h1 , . . , hr } be a basis for W ⊥ . Extend this to a basis for the whole space, {h1 , . . , hr , f 1 , . . , f n−r }. Every vector v in the column space of G is expressible as v = G BT where b is a linear combination of the basis vectors {h1 , . . , hr , f 1 , . . , f n−r }. Hence every vector v in the column space of G must be expressible as a linear combination of {GhT1 , GhT2 , . .

An ). Hence we need consider only vector spaces of n-tuples. The inner product of two n-tuples of F n u = (a1 , . . , an ) v = (b1 , . . , bn ) is defined as the scalar u · v = (a1 , . . , an ) · (b1 , . . , bn ) = a1 b1 + · · · + an bn . 36 Introduction to Algebra We can verify immediately that u · v = v · u, that (cu) · v = c(u · v), and also that w · (u + v) = (w · u) + (w · v). If the inner product of two vectors is zero, they are said to be orthogonal. It should be noted that it is possible for a nonzero vector over G F(q) to be orthogonal to itself.

3 because a −1 does not exist in this ring. There is a special name for rings in which cancellation is always possible – a domain. 4. A domain (or integral domain) is a commutative ring in which b = c whenever ab = ac and a is nonzero. 5 Vector spaces A familiar example of a vector space is the three-dimensional euclidean space that arises in problems of physics. This can be extended mathematically to an n-dimensional vector space over the real numbers. The concept of an n-dimensional vector space is closely related to the ideas of linear algebra and matrix theory, and is important in many applications.

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