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Unformatted text preview: EE 387, John Gill, Stanford University Notes #5, October 31, Handout #21 Cyclic codes: overview A linear block code is called cyclic if every cyclic shift of a codeword is a codeword. Cyclic codes have many advantages. Elegant algebraic descriptions: c ( x ) = m ( x ) g ( x ) , where g ( x ) is generator polynomial c ( x ) h ( x ) = 0 mod ( x n 1) , where h ( x ) is parity-check polynomial c ( 1 ) = 0 ,...,c ( t ) = 0 , where i GF( q m ) Shift register encoders and syndrome units Simple burst error correction ( error trapping ) Random error correction by solving polynomial equations Cyclic codes are (by definition) a very special case of linear codes. But nearly all commonly used block codes are (shortened) cyclic codes. EE 387 Notes #5, Page 1 Cyclic shifts of n-tuples and polynomials Definition : The right cyclic shift of ( v ,v 1 ,...,v n 1 ) is ( v n 1 ,v ,...,v n 2 ) . CLK D Q D Q D Q D Q D Q D Q D Q The right cyclic shift of v is denoted by v (1) . When n-tuples are polynomials of degree n 1 , the right cyclic shift of v ( x ) = ( v ,v 1 ,...,v n 2 ,v n 1 ) = v + v 1 x + + v n 2 x n 2 + v n 1 x n 1 is also a polynomial of degree n 1 : v (1) ( x ) = ( v n 1 ,v ,v 1 ,...,v n 2 ) = v n 1 + v x + v 1 x 2 + + v n 2 x n 1 The right cyclic shift of v = v ( x ) by i positions is denoted by v ( i ) = v ( i ) ( x ) . EE 387 Notes #5, Page 2 Properties of cyclic shifts Some obvious facts about cyclic shifts of n-tuples: v ( n ) = v . Left cyclic shift i positions v ( i ) is same as right cyclic shift v ( n i ) . v ( i ) = v ( i mod n ) . The first two statements are special cases of the third. The right cyclic shift operation is a linear transformation v (1) = v S 1 , where S 1 is the permutation matrix S 1 = 1 1 . . . . . . . . . . . . . . . 1 1 . Every linear feedback shift register corresponds to a matrix in rational canonical form. The cyclic register is the simplest feedback shift register. EE 387 Notes #5, Page 3 Numbers vs. polynomials When binary n-tuples represent integers, bit shifts perform multiplication and division by powers of 2. Logical left shift is multiplication by 2. Logical right shift is unsigned division by 2. Cyclic left and rights shifts are multiplication and division by 2 mod 2 n 1 . When n-tuples represent polynomials over a field, cyclic shifts correspond to multiplication and division by powers of x mod ( x n 1) . We write polynomials with the most significant coefficients on the right ....
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This note was uploaded on 05/19/2011 for the course ECE 635 taught by Professor Profnaganagi during the Spring '09 term at CSU Northridge.
- Spring '09