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Unformatted text preview: erformed modulo 2 (it’s the same as taking the exclusive OR of the bits). Even parity
occurs when the sum is 0 (i.e., the number of 1’s is even), while odd parity is when the
sum is 1.
Parity, or equivalently, arithmetic modulo 2, has a special name: algebra in a Galois
Field of order 2, also denoted F2 . A ﬁeld must deﬁne rules for addition and multiplication.
Addition in F2 is as stated above: 0 + 0 = 1 + 1 = 0; 1 + 0 = 0 + 1 = 1. Multiplication is as
usual: 0 · 0 = 0 · 1 = 1 · 0 = 0; 1 · 1 = 1. Our focus in 6.02 will be on linear codes over F2 ,
but there are natural generalizations to ﬁelds of higher order (in particular, Reed Solomon
codes, which are over Galois Fields of order 2q ).
A linear block code is characterized by the following rule (which is both a necessary
and a sufﬁcient condition for a code to be a linear block code):
Deﬁnition 6.1 A block code is said to be linear if, and only if, the sum of any two code words is
another code word.
For example, the code deﬁned by code words 000, 101, 011 is not a l...
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This document was uploaded on 02/26/2014 for the course CS 6.02 at MIT.
 Fall '13
 HariBalakrishnan

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