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Uniquely Decipherable Codes: Definition and McMillan's Theorem

This lecture covers the definition of uniquely decipherable codes, which are codes where no two different sequences of codewords form the same string. It also introduces McMillan's Theorem, which provides a necessary condition for unique decipherability. The lecture includes examples and homework exercises for further practice.

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Uniquely Decipherable Codes: Definition and McMillan's Theorem

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  1. Lecture 5Uniquely Decipherable Codes(Section 1.3) Theory of Information

  2. Definition of Unique Decipherability DEFINITION A code C over an alphabet A is uniquely deciphereable iff no two different sequences of codewords form the same string over A. I.e., for any c1,…,cn,d1,…,dmC, we have: if c1c2…cn=d1d2…dm, then m=n and c1=d1, c2=d2, …,cn=dn. Example 1.3.1. C={c1=0, c2=01, c3=001} D={d1=0, d2=10, d3=110} Read in C: 00 010 1000 010100 001 Read in D: 00 010 1000 010100 001 C is not uniquely decipherable (because of 001) D is uniquely decipherable (because no codeword is a prefix of another codeword)

  3. McMillan’s Theorem THEOREM 1.3.1. Let C={c1,…,cq} be an r-ary code. If C is uniquely decipherable, then 1/rlen(c1)+…+1/rlen(cq) 1. Note: This theorem only establishes a necessary but not sufficient condition for unique decipherability. Does C={c1=0, c2=01, c3=001} pass McMillan’s test? Can {0, 10, 110, 111,101} be uniquely decipherable?

  4. Homework Exercises 1,2,3,4,5,6,7,8 of Section 1.3.

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