3/7/20171Ma/CS 6bClass 25: Error Correcting Codes 2By Adam ShefferRecall: Codes??–the set of binary sequences of length ?.◦For example, ?3=000,001,010,011,100,101,110,111 .Codesof 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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3/7/20172Recall: DistancesGiven ?, ? ∈ ??the distance𝜕?,?is the number of bits that are different in ?and ?.◦𝜕010101,000111=2.Given a code 𝐶 ⊂ ??, the distance of 𝐶is min?,?∈𝐶𝜕?,?.If 𝐶has distance ?:◦We can detectany errors, as long as no more than ?−1bits were flipped.◦We can correctany errors, as long as fewer than ?/2bits were flipped.Recall: Linear Codes A code 𝐶 ⊂ ??is linearif for every ?, ? ∈ 𝐶, we have ? + ? ∈ 𝐶.For any ?×?matrix ?, the set 𝐶 =? ∈ ??: ?? = 0?is a linear code. We say that ?is the check matrixof 𝐶.Theorem.If no column in ?consists entirely of zeros and no two columns of ?are identical, then 𝐶can correct at least one bit flip.
3/7/20173Code SizeProblem.We are interested in a code with a check matrix ?that has 𝑟rows and does not contain zero columns and identical columns. What is the maximum number of columns in ??◦There are 2𝑟distinct columns of 𝑟entries. ◦After ignoring the all zeros column, we remain with 2𝑟−1possible columns.◦For example, when 𝑟 = 3, we have100110101010110010111.Recall: Standard FormWe consider matrices of the form?=100⋯0?1,1⋯?1,?−𝑟010⋯0?2,1⋯?2,?−𝑟0⋮00⋮01⋮0…⋱⋯0⋮1?3,1⋮?𝑟,1⋯?3,?−𝑟⋱⋮⋯?𝑟,?−𝑟.This is called standard form.