Cyclic Codes

2. Cyclic Codes • Hamming code is useful but there exist codes that offers same (if not larger) error control capabilities while can be implemented much simpler. • Cyclic code is a linear code that any cyclic shift of a codeword is still a codeword. • Makes encoding/decoding much simpler, no need of matrix multiplication.

3. Cyclic code • Polynomial representation of cyclic codes, here, assume all coefficients are either 0 or 1. • That is, if your code is (1010011) (c6 first, c0 last), you write it as • Addition and subtraction of polynomials --- Done by doing binary addition or subtraction on each bit individually, no carry and no borrow. • Division and multiplication of polynomials. Try divide by .

4. Cyclic Code • A (n,k) cyclic code can be generated by a polynomial g(x) which has degree n-k and is a factor of xn-1. Call it the generator polynomial. • Given message bits, (mk-1, …, m1, m0), the code is generated simply as: • In other words, C(x) can be considered as the product of m(x) and g(x).

5. Example • A (7,4) cyclic code g(x) = x3+x+1. • If m(x) = x3+1, C(x) = x6+x4+x+1.

6. Cyclic Code • One way of thinking it is to write it out as the generator matrix • So, clearly, it is a linear code. Each row of the generator matrix is just a shifted version of the first row. Unlike Hamming Code. • Why is it a cyclic code?

7. Example • The cyclic shift of C(x) = x6+x4+x+1 is C1(x) = x5+x2+x+1. • It is still a code polynomial, because it the code polynomial if m(x) = x2+1.

8. Cyclic Code • Given a code polynomial • We have • C1(x) is the cyclic shift of C(x) and (1) has a degree of no more than n-1 and (2) divides g(x) (why?) hence is a code polynomial.

9. Cyclic Code • So, to generate a cyclic code is to find a polynomial that (1) has degree n-k and (2) is a factor of xn-1.

10. Generating Systematic Cyclic Code • A systematic code means that the first k bits are the data bits and the rest n-k bits are parity checking bits. • To generate it, you let where • The claim is that C(x) must divide g(x) hence is a code polynomial. 33 mod 7 = 5. Hence 33-5=28 can be divided by 7.

11. Division Circuit • Division of polynomials can be done efficiently by the division circuit. (just to know there exists such a thing, no need to understand it)

12. Remaining Questions for Those Really Interested • Decoding. Divide the received polynomial by g(x). If there is no error you should get a 0 (why?). Make sure that the error polynomial you have in mind does not divide g(x). • How to make sure to choose a good g(x) to make the minimum degree larger? Turns out to learn this you have to study more – it’s the BCH code.

13. Finite Field • A field is a set of elements plus the + and * operation:

14. Finite Field • Continued…

15. Finite field • A finite field is a field with finite number of elements. • The simplest field is the binary field: {0,1}. The + operation is bitwise xor and the * operation is bitwise and. • All requirements satisfied.

16. Beyond finite field • Why do we have to deal with fields other than the binary field? • Better code can be constructed. • We only consider the extension field of the binary field. • GF(2m) contains 2m elements.

17. GF(2m) • It is convenient to represent the 2m elements in GF(2m) as polynomials. • The difference with the cyclic code is that here the polynomial represent the element, not a codeword! • The 8 element in GF(23) can be represented as:

18. GF(23) • represents ithpolynomial in GF(23). • The + operation is defined as adding two polynomials: • For example,

19. GF(23) • The * operation is defined as where f(y) has degree 3 and is irreducible (cannot be expressed as the product of two polynomials )and is called the “base polynomial.” • For example, f(y)=y3+y+1.

20. GF(23) • 0 and 1 element in GF(23) are just 0 and 1, respectively. • As long as f(y) is irreducible, other requirements all satisfy. • For example, the reciprocal of an element. An element multiplying all other elements must result in different results if f(y) is irreducible.

21. Primitive element • Consider the power series an element: • Because the number of elements is finite, at some point, it will repeat a previous element. • We will be able to find , and therefore and we say it has order n. • It can be shown that there exists an element with order 2m-1. • This element is called the primitive element and denoted as

22. Primitive element • The powers of the primitive element generates all elements. Let • Non-primitive elements may have order less than 2m-1, but are all factors of it.

23. Finite Field • After defining the elements in the finite filed and the + and * operation, we can plug an element into a polynomial. • For example, if g(x) = x4+x3+x2+1 , and we substitute x with . That is, find the value of • Since the result is 0. • In this sense, is the root of g(x).

24. Minimum Polynomial • Given an element, the minimum polynomial is defined as the minimum degree polynomial such that this element is a root of it. • The minimum polynomial is irreducible.

25. BCH code • The BCH code is defined on some finite field GF(q). Meaning that the elements in the codeword must be from GF(q). • The requirement is that the length of the code n must be n=qm-1 for some m. • Given m and n, suppose is a primitive element of GF(qm) with order n. • Get d-1elements: • The generator polynomial is the least common multiple of the product of the minimum polynomials for

26. BCH code • The BCH code has minimum distance d. • Proof: Suppose the minimum distance is less than d. It means that there exists a code polynomial with less than d non-zero coefficients. Let this code polynomial be so, because are the roots of C(x), we have

27. BCH code • This means that • Or, the determinant of the matrix must be 0

28. BCH code