Theory of Information Lecture 20 Linear Codes Section

Theory of Information Lecture 20 Linear Codes (Section 5. 2) 1

Definition of Linear Codes Definition A code C Znp that is also a subspace of Znp is called a linear code. If C has dimension k and minimum distance d(C)=d, then C is an [n, k, d]-code. The numbers n, k, d are called the parameters of C. The parameter d can be omitted when irrelevant. Theorem 5. 2. 1 A p-ary linear [n, k, d]-code C has size pk and is thus an (n, pk, d)-code. Proof. Let B={x 1, …, xk} be the basis of C. Then any codeword x can be written as x = 1 x 1+…+ kxk for a unique set of scalars. Hence, there as many codewords as the number of ordered k-tuples of scalars, and this number is pk. Example 5. 2. 1 The binary (4, 4)-code C 1={0000, 1011, 0110, 1101} is a subspace of Z 42 And thus it is linear. B={1011, 0110} is a basis for it. Therefore dim(C 1)=2. Hence, C 1 is a [4, 2]-code. 2

Minimum Distance of a Linear Code Definition The weight w(c) of a word c is the number of places in c with nonzero entries. The weight w(C) of a code C is the minimum weight of all nonzero codewords in C. Theorem 5. 2. 3 If C is a linear code, then d(C)=w(C). Example 5. 2. 3 What are the minimum distances of the following linear codes? {0000, 1011, 0110, 1101} {0000, 0121, 0212, 2210, 2001, 2122, 1120, 1211, 1002} Why is Theorem 5. 2. 3 important? 3

Another Appeal of Linear Code Another big advantage of linear codes is that a p-ary linear [n, k]-code C can be described by just giving a basis for C, which only contains k words, rather than listing the pk individual codewords of C. Say, for a binary code of dimension 30, a basis has size 30 while the code size is over a billion! 4

Homework Exercises 1, 3, 8, 9. Remember (and understand) all of the items of this lecture marked as Definition or Theorem. 5