Cyclic Codes Introduction Binary cyclic codes form a subclass of linear block codes. Easier to encode and decode

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1 Cyclic Codes Introduction Binary cyclic codes form a subclass of linear block codes. Easier to encode and decode Definition A n, k linear block code C is called a cyclic code if. The sum of any two codewords in the code is also a codeword. Linear Example: C i C j C k CC.. Any cyclic shift of a codeword in the code is also a codeword. Cyclic Example: If C [ C C L C ] n is a codeword, C C M C n [ Cn C L Cn Cn ] [ C C L C C ] n are also codewords. [ C C L C C ] n n n n CC.

2 C and C i We can represent the code word C[C C C n- ] by a code polynomial C C C C... C n The coefficients C i {,} and each power of in the polynomial C represents a one-bit shift in time. Hence, multiplication of the polynomial C by may be viewed as a shift to the right. n- Example: C [] can be represented by C CC. C i is recognized as the code polynomial of the code word [C n-i C n- C C C n-i- ] obtained by applying i cyclic shifts to the code word [C C C n- ]. It can be shown that C i is the remainder resulting from i n dividing C by. That is, i C q n C i where q C n i Cn i... C n i Proof refers to p., Communication Systems CC.

3 CC.5 Example: C[] C 5 C 5 7 C [] Remainder CC. Therefore, if C is a code polynomial, then the polynomial is also a code polynomial for any cyclic shift i. modulo mod mod n i i C c

4 Generator Polynomial Theorem If g is a polynomial of degree n - k and is a factor of n, then g generates an n, k cyclic code in which the code polynomial C for a data vector M [ m m m... m k- ] is generated by C M g n where C C C C... C n M g k m m m... m k n k g g g... g n k g is the generating polynomial CC.7 Example As 7 we can use either or to generate a 7, cyclic code. For M [ ] and g. C M g C [] [b:m] not systematic CC.8

5 The remaining code vectors are Message Code vectors obtained using CMg Right-shifted bit Right-shifted 5 bit Right-shifted bit Right-shifted bit : : CC.9 Systematic cyclic code generation Suppose we are given the generator polynomial g and the requirement is to encode the message sequence m, m,..., m k into an n, k systematic cyclic code. That is, the message bits are transmitted in unaltered form, as shown by the following structure for a code word b, b,..., bn k, m, m,..., mk n k parity bits k message bits Let the message polynomial be defined by k M m m... m k and let B b b... b n-k- n-k- CC.

6 We want the code polynomial to be in the form C B n-k M Hence, Ag B n-k M Equivalently, we may write k M A g n B g This equation states that the polynomial B is the remainder left over after dividing n k M by gx. CC. We may now summarize the steps involved in the encoding procedure for an n, k cyclic code assured of a systematic structure. Specifically, we proceed as follows:. Multiply the message polynomial M by n k.. Divide n k M by the generator polynomial g, obtaining the remainder B.. Add B to n k M, obtaining the code polynomial C. CC.

7 Example Consider the 7, cyclic code in CC.8: For M [ ] and g M n k 5 M. The division of 5 by g. can be done as CC. 5 5 Subtraction is the same as addition in modulo- arithmetic Hence, n k B and then C B M C [] 5 CC.

8 Parity-check polynomial An n,k cyclic code is uniquely specified by its generator polynomial g of order n-k. Such a code is also uniquely specified by another polynomial of degree k, which is called the parity-check polynomial, defined by k k h h h... hk. In linear block code, we have GH T. Now, we have n g h mod and we state that n g h CC.5 Syndrome Let the received word be r r... R r r r... r n [ r n n ] and Now, R q g S where S is called syndrome polynomial because its coefficients make up the syndrome S. CC.

9 CC.7 Example Hamming Codes Consider the 7, cycle code Let us take as the generating polynomial. This means that and the parity-check polynomial is Take 7 g 7 h g h ] [ M M CC.8 Now M k n

10 Thus, the quotient A and the remainder B are A B The codeword is therefore A [] B [] { { parity check bit message CC.9 Let the transmitted code be [] and the received word be []. The syndrome is []. Since the syndrome is nonzero, the received word is in error. Using the standard array, we see that the error pattern corresponding to this syndrome is []. This indicates that the error is in the middle bit of the received word. CC.

11 Cyclic Redundancy Check Codes CRC. CRC can designed to detect many combinations of likely errors. Example: error burst contiguous error bits C[ ] R[ ] CC.. The implementation of both encoding and error-detecting circuits is practical. Flip-flops and adders n k Example: n,k g g g L g n k Encoder: Parity bits Gate g g... g n-k- adder Flip-flop Syndrome calculator:... Message bits Gate g g... g n-k-... Flip-flop adder CC.

12 Binary n,k CRC codes are capable of detecting the following error patterns: a. All error bursts of length n - k or less. b. A fraction of error bursts of length equal to n - k ; the fraction equals - -n-k-. c. A fraction of error bursts of length greater than n - k ; the fraction equals - -n-k-. d. All combinations of d min - or fewer errors. e. All error patterns with an odd number of errors if the generator polynomial g for the code has an even number of nonzero coefficients. CC. Examples Code g n-k CRC CRC- CRC- USA 5 CRC-ITU 5 CRC- code is used for -bit characters transmission. CRC- and CRC-ITU are used for -bit characters transmission. CC.

13 Bose-Chaudhuri-Hocquenghem BCH codes Primitive BCH codes are characterized for any positive integer m equal or greater than and t less than m -/ by the following parameters: Block length: n m - Number of message bits: k n - mt Minimum distance: d min t Hamming single-error correcting codes is a BCH code. CC.5 Reed-Solomon Codes RS An important subclass of nonbinary BCH codes. RS code differs from a binary encoder is that it operates on multiple bits rather than individual bits. An RS n,k code is used to encode k m-bit symbols into blocks consisting of n m - symbols. Used in compact disc digital audio systems CC.