25 Codes 2.pdf - Ma/CS 6b Class 25 Error Correcting Codes 2 By Adam Sheffer Recall Codes – the set of binary sequences of length ◦ For example 3 =

25 Codes 2.pdf - Ma/CS 6b Class 25 Error Correcting Codes 2...

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3/7/2017 1 Ma/ CS 6 b Class 25: Error Correcting Codes 2 By Adam Sheffer Recall: Codes ? ? the set of binary sequences of length ? . For example, ? 3 = 000,001,010,011,100,101,110,111 . Codes of length ? are subsets of ? ? . For example, we might have 000 = up, 110 = down, 011 = left, 101 = right . This code is 000,110,011,101 ⊂ ? 3 . Sequences that are in the code are called codewords (or just words ).
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3/7/2017 2 Recall: Distances Given ?, ? ∈ ? ? the distance 𝜕 ? , ? is the number of bits that are different in ? and ? . 𝜕 010101,000111 = 2 . Given a code 𝐶 ⊂ ? ? , the distance of 𝐶 is min ? , ? 𝐶 𝜕 ? , ? . If 𝐶 has distance ? : We can detect any errors, as long as no more than ? 1 bits were flipped. We can correct any errors, as long as fewer than ? / 2 bits were flipped. Recall: Linear Codes A code 𝐶 ⊂ ? ? is linear if for every ?, ? ∈ 𝐶 , we have ? + ? ∈ 𝐶 . For any ? × ? matrix ? , the set 𝐶 = ? ∈ ? ? : ?? = 0 ? is a linear code . We say that ? is the check matrix of 𝐶 . Theorem. If no column in ? consists entirely of zeros and no two columns of ? are identical , then 𝐶 can correct at least one bit flip.
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3/7/2017 3 Code Size Problem. We are interested in a code with a check matrix ? that has 𝑟 rows and does not contain zero columns and identical columns . What is the maximum number of columns in ? ? There are 2 𝑟 distinct columns of 𝑟 entries. After ignoring the all zeros column, we remain with 2 𝑟 1 possible columns. For example, when 𝑟 = 3 , we have 1 0 0 1 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1 . Recall: Standard Form We consider matrices of the form ? = 1 0 0 0 ? 1 , 1 ? 1 , ? 𝑟 0 1 0 0 ? 2 , 1 ? 2 , ? 𝑟 0 0 0 0 1 0 0 1 ? 3 , 1 ? 𝑟 , 1 ? 3 , ? 𝑟 ? 𝑟 , ? 𝑟 . This is called standard form .
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