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De nition 2.9 is also nonsingular. Any nonsingular matrix can be transformed into an identity matrix by elementary row operations. It can be proved that if M is an n m matrix and S is a nonsingular n n matrix, then the product of S and M has the same row space as M has. The nonsingular matrices will be extensively used for designing matrix codes in later chapters. Transposed Matrix The transpose of an n m matrix M is an m n matrix, denoted MT , whose rows are the columns of M and thus whose columns are the rows of M. Vandermonde Matrix The following square matrix with elements ai s from the nite or in nite eld is called a Vandermonde matrix, and its determinant is nonzero if the elements ai s are distinct: 2 6 6 6 6 6 4 1 a1 a1 2 . . . a1 n 1 1 a2 a2 2 . . . a2 n 1 1 a3 a3 2 . . . a3 n 1 1 an an 2 . . . 3 7 7 7 7 7 5
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Since the determinant of this square matrix with distinct ai s is nonzero, the matrix is nonsingular. This matrix will be used again in later chapters, in particular, in 7. 2.2.4 Distance and Error Control Capability
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We begin with the important concepts of Hamming weight and Hamming distance. De nition 2.19 The Hamming weight of a vector u u0 ; u1 ; . . . ; un 1 , denoted by w u , is the number of nonzero elements of u. & De nition 2.20 The Hamming distance between two vectors u and v, denoted by d u; v , is the Hamming weight of u v. The Hamming distance also equals the number of positions by which the two vectors differ. That is, d u; v w u v w v u number of differing positions of u and v: & The Hamming distance is a metric in the sense that it is a real number satisfying the following: (1) d u; v > 0 for u 6 v positive definiteness 0 for u v (2) d u; v d v; u symmetry (3) d u; v d v; w ! d u; w triangle inequality . The Hamming distance and the Hamming weight can often help us understand the error control capability of a code. When the codeword v is transmitted, and hence we receive the 2:2
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erroneous word r, the Hamming distance between v and r, meaning d v; r , is equal to the number of errors. The Hamming weight of the error vector e r v, meaning w e , provides the number of errors. De nition 2.21 The minimum Hamming distance (or minimum distance) dmin of a code C is the minimum of the distances between all pairs of codewords. & Since vectors u and v are codewords in a linear code in Eq. (2.2), then u v or v u is also another codeword. Therefore the minimum distance of a linear code is equal to the minimum weight of its nonzero codewords. The minimum distance of a code is an important parameter by which we decide the error control capability of the code. As was mentioned before, every linear code contains a zero codeword. Now we consider V, the set of all n-tuples over GF 2 , and let a subset C & V be a code with minimum distance d. The codeword is used as a transmitted word. At the receiver, the received word is checked to see whether or not it is a codeword. If it is a codeword, then it is accepted; otherwise, it is considered to be an erroneous word, and the errors are detected. Let us consider the relation between the minimum distance and the code capability. Error Detection If the code has the minimum distance at least d, then the code detects any error pattern of weight d 1 or less. No pattern of d 1 or fewer errors can change the transmitted codeword into another codeword. This is because dmin is d. Therefore such errors can be detected. This means that the code with dmin d guarantees detection of d 1 or fewer errors. Error Correction If the code has the minimum distance dmin larger than or equal to 2t 1, then the code can correct all patterns of t or fewer errors. Here we consider the n-dimensional spheres of radius t for each codeword as its center. These spheres are all disjoint, and for any t or fewer errors from a codeword, the erroneous words are present within the bounds of the respective sphere. On the other hand, if the minimum distance is less than 2t 1, the t-error patterns have cases to result in a received word at least as close to an incorrect codeword as it is to the correct codeword. From these, as far as any erroneous words are present within the bounds of the sphere, these can be recovered to the correct words. Error Correction and Detection If the minimum distance of the code dmin is larger than or equal to t d 1, then the code can correct any combination of t errors and detect up to d errors where d is larger than or equal to t. We consider again spheres of radius t from each codeword as its center. For a codeword, t or fewer errors cause the received word to fall within its own sphere. That is, these errors can be corrected. Any errors larger than t but less than d ! t apart from the center of the spheres can be detected because the received words are outside all these spheres. The spheres are basically all disjoint, because if there is an intersection containing a word, then this word is, at most, distance t from two codewords, whereas the two codewords are apart by more than 2t 1. This violates triangle inequality, and is therefore impossible. Similarly, if the number of errors are larger than t but not larger than d, then the received word is outside these spheres, so the errors can be detected but not corrected. The above relations between the minimum distance and the code capability are illustrated in Figure 2.1.
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