2001 년 06 월 1 일 김 대 선 연세대학교 전기전자공학과

1. M-sequences Decimation of m-sequences 2001년 06월 1일 김 대 선 연세대학교 전기전자공학과 <유한체 이론 및 응용>

2. Table of Contents • Introduction Decimation of m-sequences • Theorem 10.9 • There exists m-sequences having same length n • Corollary 10.10 • We can obtain any m-sequences of length n by suitable decimation • Example 10.3 • Set of any number for decimation <유한체 이론 및 응용>

3. Introduction Decimation of m-sequences • If we are given just one m-sequences of length , how could find other ones of length => We can use the notion of decimation => We will prove next theorem and corollary • A sequence , and any integer => a dth decimation of is any sequence obtained by taking every dth term of the original seqeunce <유한체 이론 및 응용>

4. Theorem 10. 9 • If and : two m-sequences of length ,satisfy th10.7 => there exists an integer d, relatively prime to such that pf) By Theorem 10.7 By Lemma 5.4 ( if then ) there exists with such that <유한체 이론 및 응용>

5. Corollary 10.10 • Any m-sequence of length can be obtained from any other by suitable decimation pf) , : any two m-sequences of length By Theorem 10.6 for where and <유한체 이론 및 응용>

6. By Theorem 10.9 for some integer d can be obtained by decimating by d , starting with the term <유한체 이론 및 응용>

7. Example 10.3 • Consider the m-sequences of length 7 a root of the primitive polynomial 1 0 0 1 0 1 1 1 0 0 1 0 1 1 … If decimate by d=2 1 0 0 1 0 1 1 … From theorem 10.7 <유한체 이론 및 응용>

8. and similarly d=4, 8, 16, etc. If d=3 1 1 1 0 1 0 0… => different m-sequences characteristic polynomial and d=5, d=6 same because , conjugate of <유한체 이론 및 응용>

9. Set of any number for Decimation • We can decimate it by any number d lying in the set and get another m-sequences the subset of the same as the original • : a multiplicative group : a subgroup <유한체 이론 및 응용>

10. Example m = 6, the group has the cosets of with respect to We get all 6 cyclically distinct m-sequences of length 63 <유한체 이론 및 응용>