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De nition 2.13 A set of vectors is linearly independent if it is not linearly dependent. That is, c1 v1 c2 v2 ck vk 6 0. & In such a set, any one of those cannot be obtained as a linear combination of the others. De nition 2.14 A set of vectors v1 , v2 ; . . . ; vk , is said to span a vector space V if and only if every vector in V equals a linear combination of the vectors in the set. & In this case, if a set of vectors v1 , v2 ; . . . ; vk , span V, then any set that is linearly independent in V can have at most k vectors. Readers are recommended to prove this. This states that there can be at most k linearly independent vectors in a vector space that is spanned by k vectors. Therefore, if k linearly independent vectors v1 , v2 ; . . . ; vk span V, then k is said to be the dimension of V and the set fv1 ; v2 ; . . . ; vk g is a basis of V. Also in a k-dimensional space any basis must be exactly of k vectors, and the vectors are linearly independent. If V is any k-dimensional vector space, then any set of k linearly independent vectors is a basis of V. Orthogonality and Null Spaces De nition 2.15 An inner product or dot product of two k-tuple vectors u and v is a scalar and is de ned as follows: u v u1 u2 . . . uk v1 v2 . . . vk u1 v1 u2 v2 uk vk : & It is easily veri ed that u v v u and that w u v w u w v for any k-tuple vectors u, v, and w. De nition 2.16 If the inner product of two vectors u and v is zero, that is, u v v u 0, then they are said to be orthogonal. & Let S1 be a k-dimensional subspace of V n , where V n is the vector space of all n-tuple vectors, such that for any u 2 S2, u is orthogonal to every vector in S1 . Then the set of vectors S2 orthogonal to every vector in subspace S1 of the vector space V n is also a subspace. In this case S2 is called the null space of S1 . If S2 is the null space of S1 , then it is easy to see that S1 is the null space of S2 . The dimension of S2 is given by the following: if S2 is the null space of a subspace S1 with dimension k in V n , then S2 has dimension n k. 2.2.2 Linear Codes as Vector Spaces
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Let the vector space V n of all n-tuples over GF q be expressed as V n f a0 a1 a2 . . . an 1 j ai 2 GF q g:
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We de ne linear codes as follows: De nition 2.17 subspace. A subset C of V n is a linear code over GF q if and only if it is a &
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That is, for all w0 , w1 2 C, and for all c0 ; c1 2 GF q if c0 w 0 c1 w 1 is also included in C, then C is called a linear code over GF q . For q 2, a word 0 0; 0; . . . ; 0 is a codeword in C over GF 2 . Also, if w1 w2 is a codeword for all w1 , w2 2 C, then C is a linear code. As a simple example, set of f 0 0 0 , 0 1 1 , 1 0 1 , 1 1 0 g is a linear code because 0 0 0 0 is included in the set, and sum of any two codewords is also a codeword. Using k codewords w0 , w1 ; . . . ; wk 1 in C, a linear combination of these, that is, w c0 w0 c1 w1 ck 1 wk 1 ; c0 ; c1 ; . . . ; ck 1 2 GF q ; 2:1
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is a codeword of C over GF q from the de nition of the linear code. Here we assume that w0 , w1 ; . . . ; wk 1 , are linearly independent over GF q . Then, if any codeword of C can be expressed by Eq. (2.1), we call fw0 , w1 ; . . . ; wk 1 g a basis of the linear code C. The number of codewords k in the basis is called a dimension of the linear code C, denoted dim C . In the simple example above, there are three kinds of basis, f 0 1 1 , 1 0 1 g, f 0 1 1 , 1 1 0 g, f 1 0 1 , 1 1 0 g, and dim C 2. It can be proved that since w0 , w1 ; . . . ; wk 1 , are linearly independent, the coef cients c0 ; c1 ; . . . ; ck 1 , are uniquely determined for each codeword. Therefore the q-ary linear code with dimension dim C k has qk distinct codewords. 2.2.3 Matrix Algebra
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Basic knowledge of matrix algebra is important for designing matrix codes. This subsection presents a brief introduction to matrix algebra, and includes the important matrices that will be applied to block codes in later chapters. An n m matrix is an ordered set of nm elements over GF q in a rectangular array of n rows and m columns: 3 2 a1;1 a1;2 a1;m 6 a2;1 a2;2 a2;m 7 7 6 6 . . 7 ai;j ; .. . . 5 . 4 . . . . . an;1 an;2 an;m where i 1; 2; . . . ; n, and j 1; 2; . . . ; m. The n rows can be thought of as n m-tuples or vectors, and similarly, the m columns can be thought of as m n-tuples or vectors. The set of elements ai; j for which the column number and row number are equal is called the main diagonal. The row space of an n m matrix is the set of all linear combinations of row vectors of the matrix. They form a subspace of the vector space of m-tuples. The dimension of the row space is called the row rank. Similarly the set of all linear combinations of column vectors of the matrix forms the column space, whose dimension is called the column rank. It can be shown that if row rank equals column rank, then this is referred to as rank of the matrix.
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