Information Theory. Lecture 7


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1 Information Theory Lecture 7 Finite fields continued: R3 and R7 the field GF(p m ),... Cyclic Codes Intro. to cyclic codes: R8.1 3 Mikael Skoglund, Information Theory 1/17 The Field GF(p m ) π(x) irreducible degreem over GF(p), p a prime, GF(p m ) = all polynomials over GF(p) of degree m 1, with calculations modulo p and π(x) modulo π(x) use π(x) = 0 to reduce x m to degree < m without loss of generality, π(x) can be assumed monic The prime number p is called the characteristic of GF(p m ); smallest p such that p i=1 1 = 0 GF(p m ) is a linear vector space of dimension m over GF(p) For s < r, GF(p s ) GF(p r ) s r For β GF(p r ), β GF(p s ) β ps = β Mikael Skoglund, Information Theory 2/17
2 The Cyclic Group G = GF(p m ) \ {0} For any β GF(p m ), the smallest r > 0 such that β r = 1 is called the order of β. The elements in G = GF(p m ) \ {0} form a cyclic group; There exists an element α GF(p m ) of order r = p m 1 that generates all the nonzero elements of GF(p m ), that is G = {1, α, α 2,..., α r 1 } Any such α is called a primitive element = Fermat s theorem: Any β GF(q) satisfies β q = β, that is x q x = β GF(q) (x β) = x r 1 (x α i ) i=0 Mikael Skoglund, Information Theory 3/17 Polynomial Factorizations For β GF(p m ) the minimal polynomial of β is the lowest degree monic polynomial m(x) over GF(p) with β as a root m(x) is irreducible, has degree s m such that s m, and roots β, β p, β 2p,..., β (s 1)p called conjugates If f(β) = 0 for f(x) m(x) over GF(p), then m(x) f(x); f(β) = 0 = f(β p ) = 0 The minimal polynomial of a primitive element in GF(p m ) has degree m, and is called a primitive polynomial Mikael Skoglund, Information Theory 4/17
3 A field has at least one primitive element. When generating GF(p m ) using π(x) with roots α, α p,..., α (m 1)p, the element α is primitive in GF(p m ); this is our standard primitive element, henceforth denoted α Let m (i) (x) be the minimal polynomial of α i GF(q), then x q 1 1 = t m (t) (x) over all t {1, 2,..., q 1} that give different m (t) (x) s An independent statement is: x pm x = product of all monic irreducible polynomials over GF(p) with degrees that divide m = help to identify the m (i) (x) s m (i) (x) of degree s = m ( i) (x) = x s m (i) (x 1 ) Mikael Skoglund, Information Theory 5/17 Cyclic Codes C over GF(q) is cyclic C is linear and (c 0,..., c n 1 ) C = (c n 1, c 0,..., c n 2 ) C For a cyclic code C, let c = (c 0,..., c n 1 ) C correspond to a codeword polynomial c(x) over GF(q), such that c(x) = c 0 + c 1 x + c 2 x c n 1 x n 1 A cyclic shift multiplication with x modulo x n 1 Mikael Skoglund, Information Theory 6/17
4 Formalizing... Equivalence relations: Let be a relation between objects in a set S, such that 1 x x, x S 2 x y = y x, x, y S 3 x y and y z = x z, x, y, z S A generalization of ordinary equality = Example: S = the integers, and x y if x = y modulo p Equivalence classes: An equivalence relation partitions S into elements that are equal or equivalent in the sense of, Example: 1, 8, 15,... are in the same equivalence class when = modulo 7 over the nonnegative integers Mikael Skoglund, Information Theory 7/17 Modulo a polynomial: Two polynomials a(x) and b(x) over GF(q) are equal modulo a polynomial p(x) if a(x) = q 1 (x)p(x) + r(x), b(x) = q 2 (x)p(x) + r(x) Corresponds to an equivalence relation, and a(x) and b(x) are in the same equivalence or residue class, a(x) = b(x) modulo p(x) a(x) and b(x) in the same residue class modulo p(x) Formalizing the representation of GF(p m ); let F (x) = all polynomials over a field F, take F = GF(p) and π(x) F (x) monic irreducible degreem, then GF(p m ) = F (x)/π(x) = the different residue classes of F (x) modulo π(x) Mikael Skoglund, Information Theory 8/17
5 Rings and Polynomials Let R be an Abelian group with operation + (addition), and define an operation (multiplication) such that a b = b a, a (b + c) = a b + a c, a (b c) = (a b) c for all a, b, c R, and a multiplicative identity 1 such that 1 a = a 1 = a for any a R. Then R is a ring. Mikael Skoglund, Information Theory 9/17 Let F = GF(q), and define the ring R n = F (x)/(x n 1) Each class in R n is represented by its lowest degree polynomial (of degree n 1). = in R n = mod x n 1 in the same class as the class representative = the representative R n is a linear vector space of dimension n over GF(q). A (principal) ideal g(x) R n generated by g(x) R n, g(x) = {c(x) : c(x) = u(x)g(x), u(x) R n } A cyclic code of length n with generator polynomial g(x) R n is defined as C = g(x) Mikael Skoglund, Information Theory 10/17
6 The Generator Polynomial g(x) For C = g(x), g(x) is the unique monic polynomial in C of minimal degree r the dimension of C is k = n r g(x) x n 1 any u(x) over GF(q) of degree < n r corresponds uniquely to a c(x) C via c(x) = u(x)g(x) over GF(q) k message symbols (u 0,..., u k 1 ), u l GF(q), give a codeword c(x) as c(x) = u(x)g(x), u(x) = u 0 + u 1 x + + u k 1 x k 1 C.f., c C c = ug Mikael Skoglund, Information Theory 11/17 The Parity Check Polynomial h(x) The polynomial h(x) = xn 1 g(x) is the parity check polynomial of the cyclic code g(x) of length n g(x)h(x) = 0, and c(x) g(x) c(x)h(x) = 0 in R n ; c.f., GH T = 0 and, c C ch T = 0 h(x) has degree k = dimension of g(x) Mikael Skoglund, Information Theory 12/17
7 G and H matrices For a cyclic code with g(x) = g r x r + g r 1 x r g 0 h(x) = h k x k + h k 1 x k h 0 we get G and H in cyclic form as g 0 g 1 g r G = 0 g 0 g 1 g r g 0 g 1 g r H = h k h k 1 h h k h k 1 h 0 0 h k h k 1 h Mikael Skoglund, Information Theory 13/17 Why Polynomials? Encoding and decoding circuitry based on simple logical operations straightforward to derive... Construct and analyze (cyclic) codes based on finite field theory and polynomial factorizations Mikael Skoglund, Information Theory 14/17
8 Factors of x n 1 Cyclic code over GF(q): g(x)h(x) = x n 1 = { irreducible factors } = code can be constructed based on the factors Assume (always) n and q relatively prime (no common factors) = exists a smallest m such that n q m 1 The n zeros of x n 1 GF(q m ) and no smaller field, x n 1 = n (x α i ) i=1 for some {α 1,..., α n } GF(q m ) with the α i s distinct The nth roots of unity; GF(q m ) is the splitting field of x n 1 Mikael Skoglund, Information Theory 15/17 The roots {α 1,..., α n } form a cyclic group GF(q m ), that is, there is an α GF(q m ), the primitive nth root of unity, such that n 1 x n 1 = (x α i ) i=0 n = q m 1 α is a primitive element in GF(q m ) Assume α a primitive nth root of unity GF(q m ) where m is the smallest integer such that n q m 1, p (i) (x) = minimal polynomial of α i GF(q m ) = x n 1 = j p (j) (x) over all j {0,..., n 1} that give different p (j) (x) s Mikael Skoglund, Information Theory 16/17
9 Given a factorization x n 1 = j p (j) (x) some of the p (j) (x) s can form g(x) and the others h(x); The zeros of a code, let C = g(x) of length n, and let K = {k : p (k) (x) g(x)}, then {α k : k K} are called the zeros of the code; i.e., all roots of g(x) α i for i / K (i n 1) are the nonzeros (all roots of h(x)) the nonzeros of C are the zeros of C and vice versa Mikael Skoglund, Information Theory 17/17
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