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Unformatted text preview: 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 is a primitive n th root of unity over the field GF( q ), then may or may not belong to GF( q ). If 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- )( x- 2 ) ( x- n- 1 ). This happens, for instance, if 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 = [ x ], then is a primitive 15th root of unity. A cyclic group of order 15 has (15) = 8 generators. The generators for h i are , 2 , 4 = +1, 7 = 3 + +1, 8 = 2 + 1, 11 = 3 + 2 + , 13 = 3 + 2 + 1, and 14 = 3 + 1. The minimal polynomial for over Z 2 is x 4 + x + 1, while the minimal polynomial for over GF(16) is just x- . Definition. A cyclic code of length n over GF( q ) is called a BCH code of designed distance if its generator polynomial g ( x ) GF( q )[ x ] is the least common multiple of the minimal polynomials of ` , ` +1 , . . . , ` + - 2 , where 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 . If ` = 1 in the definition of a BCH code, then it is called a narrowsense BCH code. If n = q m- 1, then the code is called a primitive BCH code (since this means that is a primitive element of GF( q m )). 1 The construction of BCH codes These codes are cyclic codes, so the construction involves finding an ideal h [ g ( x )] i in F [ x ] / h x n- 1 i , 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 ....
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