1. 2001 년 6 월 2 일 정지욱 연세대학교 전기전자공학과 Span Property 정지욱 1/7

2. Introduction to M-sequence(1) Definition An m-sequence is a binary sequence that satisfies a linear recurrence whose characteristics polynomial is primitive. (m-sequence is perhaps the best-known family of pseudo noise sequences. It is a linear feedback shift-register sequence having the maximum possible period.) Usefulness - easily generated binary sequences that behave in many respects as if they were completely random.(pseudo randomness property) - Applications: telecommunications, computer science, etc. 정지욱 2/7

3. Introduction to M-sequence(2) Period of m-sequence The characteristic polynomial of m-sequence is an irreducible polynomial of degree m which is the minimal polynomial of a primitive root in GF(2 m ). Since the characteristic polynomial of an m-sequence has period 2 m -1, every m-sequence 2 m -1.(by Theorem 9.4) Example 10.1 primitive polynomial: 정지욱 3/7

4. Example of m-sequence Period: 2 4 -1=15 Initial condition: 0001 => 000100110101111 정지욱 4/7

5. Introduction to Span Property m-gram If (s 0,s 1, …., s n-1 )is the m-sequence, an m-gram is one of the n subsequences of length m of the form (s t, s t+1, ….., s t+m-1 ), for t = 0,1, …, n-1 Theorem 10.1(Span property) Among the 2 m -1 m-grams of an m-sequence {s t }, each nonzero binary vector of length m occurs once and only once. 정지욱 5/7

6. Proof of Span Property 정지욱 6/7

7. Examples of Span Property » m=3,, (s t ) = (0010111) m-grams: (001), (010), (101), (011), (111), (110), (100) » m=4,, (s t ) = (000100110101111) m-grams: (0001), (0010), (0100), (1001), (0011), (0110), (1101), (1010), (0101), (1011), (0111), (1111), (1110), (1100), (1000) » m=5,, (s t ) = (0000100101100111110001101110101) m-grams: (00001), (00010), (00100), (01001), (10010), (00101), (01011), (10110), (01100), (11001), (10011), (00111), (01111), (11111), (11110), (11100), (11000), (10001), (00011), (00110), (01101), (11011), (10111), (01110), (11101), (11010), (10101), (01010), (10100), (01000), (10000) 정지욱 7/7