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08coding

# 08coding - Math 523 Notes on Coding Theory July 2008 The...

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Math 523 Notes on Coding Theory July, 2008 The definition of BCH and RS codes Definition. A primitive n th root of unity is a root of x n - 1 that has multi- plicative order n . Example. Over the real numbers the 6th roots of unity are ± 1 , ± 1 2 ± 3 2 i . A cyclic group of order 6 has 2 generators, so there are 2 primitive 6th roots of unity: 1 2 ± 3 2 i . Note that - 1 2 ± 3 2 i are primitive third roots of unity. If β is a primitive n th root of unity over the field GF( q ), then β may or may not belong to GF( q ). If β GF( q ), then since it has order n its powers give n distinct roots of x n - 1. Therefore in this case x n - 1 factors over GF( q ) as x n - 1 = ( x - 1)( x - β )( x - β 2 ) · · · ( x - β n - 1 ). This happens, for instance, if β is an element of order p - 1 in Z p . If x n - 1 does not split in GF( q )[ x ], then to actually get our hands on a primitive n th root of unity we need to construct an extension field of GF( q ) that is a splitting field for x n - 1. More generally, we could use any extension GF( q m ) in which x n - 1 splits. Because we then have all roots of x n - 1 within the field, it follows that x n - 1 must be a factor of the polynomial x q m - x , for which GF( q m ) is a splitting field over GF( q ). Example. Over GF(2) = Z 2 , consider GF(2 4 ) = Z 2 [ x ] / x 4 + x + 1 . If we let β = [ x ], then β is a primitive 15th root of unity. A cyclic group of order 15 has ϕ (15) = 8 generators. The generators for β are β , β 2 , β 4 = β +1, β 7 = β 3 + β +1, β 8 = β 2 + 1, β 11 = β 3 + β 2 + β , β 13 = β 3 + β 2 + 1, and β 14 = β 3 + 1. The minimal polynomial for β over Z 2 is x 4 + x + 1, while the minimal polynomial for β over GF(16) is just x - β . Definition. A cyclic code of length n over GF( q ) is called a BCH code of designed distance δ if its generator polynomial g ( x ) GF( q )[ x ] is the least common multiple of the minimal polynomials of β , β +1 , . . . , β + δ - 2 , where β is a primitive n th root of unity. Definition. If n = q - 1 in the definition of a BCH code, then it is called an RS code of designed distance δ . If = 1 in the definition of a BCH code, then it is called a narrow–sense BCH code. If n = q m - 1, then the code is called a primitive BCH code (since this means that β is a primitive element of GF( q m )). 1

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The construction of BCH codes These codes are cyclic codes, so the construction involves finding an ideal [ g ( x )] in F [ x ] / x n - 1 , for some finite field F . Given g ( x ), we already know how to find a generator matrix and a parity check matrix for the code. We now outline how to construct a BCH code with designed distance δ . 1. Choose a finite field GF( q ) as the alphabet for the code. Remember that GF( q ) has characteristic p for some prime p , so q = p s for some positive integer s , and all polynomials will have coefficients from this field. 2. Choose integers n and m for which x n - 1 is a factor of x q m - x .
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08coding - Math 523 Notes on Coding Theory July 2008 The...

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