By :Er. Amit Mahajan

CYCLIC CODES

CYCLIC CODES

Definition: An (n, k) linear code C is called a cyclic code if every

cyclic shift of a code vector in C is also a code vector Codewords can be represented as polynomials of

degree n. For a cyclic code all codewords are multiple of some polynomial g(X) modulo Xn+1 such that g(X) divides Xn+1. g(X) is called the generator polynomial.

Examples: Hamming codes, Golay Codes, BCH codes, RS codes BCH codes were independently discovered by

Hocquenghem (1959) and by Bose and Chaudhuri (1960)

Reed-Solomon codes (non-binary BCH codes) were independently introduced by Reed-Solomon

Cyclic codes are of interest and importance because

• They posses rich algebraic structure that can be utilized in a variety of ways.• They have extremely concise specifications.

• They can be efficiently implemented using simple shift registers.

• Many practically important codes are cyclic.

Notations

k = Number of binary digits in the message before encoding n = Number of binary digits in the encoded message n – k = r = number of check bits

rk

n

Basic properties of Cyclic codes:

Let C be a binary (n,k) linear cyclic code1. Within the set of code polynomials in C, there is a unique monic

polynomial with minimal degree is called the generator polynomials.

2. Every code polynomial in C, can be expressed uniquely as

3. The generator polynomial is a factor of

4. The orthogonality of G and H in polynomial form is expressed as . This means is also a factor of

)(Xg )( . Xnr gr

r XgXggX ...)( 10g

)(XU

)()()( XXX gmU )(Xg

1nX

1)()( nXXX hg 1nX)(Xh

Polynomial Representation of Binary Information

It is convenient to think of binary digits as coefficients of a polynomial in the dummy variable X.Polynomial is written low-order-to-high-order.Polynomials are treated according to the laws of ordinary algebra with an exception addition is to be done modulo two.

Types of Cyclic Codes

There are two types of Cyclic Codes on basis of Encoding:

Non-Systematic Encoding: Information bits are not packed together in

the codeword. These are rarely used. c(x)= i(x) g(x) Systematic Encoding: Information bits are packed together in the

codeword. c(x)= i(x) xn-k + r(x)

8

Systematic Encoding

Systematic encoding algorithm for an (n,k) Cyclic code:

1. Multiply the message polynomial by

2. Divide the result of Step 1 by the generator polynomial . Let be the remainder.

3. Add to to form the codeword

)(Xi knX

)(Xg )(Xr

)(xiX kn

)(XC

)(Xr

Implementation

The hardware to implement this algorithm is a shift register and a collection of modulo two adders.

The number of shift register positions is equal to the degree of the divisor, G(X), and the dividend is shifted through high order first and left to right.

Circuit for encoding systematic cyclic codes

We noticed earlier that cyclic codes can be generated by using shift registers whose feedback coefficients are determined directly by the generating polynomial

Example

Decoding cyclic codes

Every valid, received code word R(p) must be a multiple of G(p), otherwise an error has occurred. (Assume that the probability of noise to convert code words to other code words is very small.)

Therefore dividing the R(p)/G(p) and considering the remainder as a syndrome can reveal if an error has happened and sometimes also to reveal in which bit (depending on code strength)

Division is accomplished by a shift registers The error syndrome of q=n-k bits is therefore

( ) mod ( ) / ( )p p pS R G

Decoding cyclic codes: error correction

Decoding circuit for (7,4) code syndrome computation

To start with, the switch is at “0” position Then shift register is stepped until all the received code bits have entered the

register This results is a 3-bit syndrome (n - k = 3 ):

that is then left to the register Then the switch is turned to the position “1” that drives the syndrome out of

the register

( ) mod ( ) / ( )p p pS R G

Decoding

Encoder and Decoder

Advantages of Cyclic codes

Simplified encoding.Easy syndrome calculation S(p) = rem{Y(p)/G(p)}Ingenious error correcting decoding methods have been devised

for specific cyclic codes. They eliminate the storage needed for table lookup.

Cyclic Redundancy Codes (CRC), a class of cyclic codes, has ability to detect burst errors of length ‘q’, all single bit errors, any odd number of errors if (p+1) is a factor of G(p), Double errors if G(p) has at least three 1’s.

Cyclic codes for error detection provides high efficiency and the ease of implementation.

It provides standardization like CRC-8 and CRC-32

Some block codes that can be realized by cyclic codes

(n,1) Repetition codes. High coding gain (minimum distance always n-1), but very low rate: 1/n

(n,k) Hamming codes. Minimum distance always 3. Thus can detect 2 errors and correct one error. n=2m-1, k = n - m,

Maximum-length codes. For every integer there exists a maximum length code (n,k) with n = 2k - 1,dmin = 2k-1.

BCH-codes. For every integer there exist a code with n = 2m-1, and where t is the error correction capability

(n,k) Reed-Solomon (RS) codes. Works with k symbols that consists of m bits that are encoded to yield code words of n symbols.

Nowadays BCH and RS are very popular due to large dmin, large number of codes, and easy generation

Code selection criteria: number of codes, correlation properties, code gain, code rate, error correction/detection properties

Reference:

1.    Communication Systems by R.P Singh , S.D Sapre 2.     Information Theory , Coding and Cryptography by Ranjan

Bose3.     http://www.cs.nmsu.edu/~pfeiffer/4.     http://web.syr.edu/~rrosenque/ecc/cyclic.htm5.      http://www.microconsultants.com/tips/crc/crc.txt