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Finite Fields: an Introduction. Part Finite fields: An introduction. Part II. Vlad Gheorghiu Department of Physics Carnegie Mellon University Pittsburgh, PA 15213, U.S.A. August 7, 2008 Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 1 / 18 Outline 1 Brief review of Part I (Jul 2, 2008) 2 Construction of finite fields Polynomials over finite fields n Explicit construction of the finite field Fq, with q = p . Examples 3 Classical coding theory Decoding methods The Coset-Leader Algorithm Examples Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 2 / 18 Examples The structure of finite fields 2 Classical codes over finite fields Linear codes: basic properties Encoding methods Hamming distance as a metric Brief review of Part I (Jul 2, 2008) Brief review of Part I 1 Finite fields Definitions Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 3 / 18 The structure of finite fields 2 Classical codes over finite fields Linear codes: basic properties Encoding methods Hamming distance as a metric Brief review of Part I (Jul 2, 2008) Brief review of Part I 1 Finite fields Definitions Examples Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 3 / 18 2 Classical codes over finite fields Linear codes: basic properties Encoding methods Hamming distance as a metric Brief review of Part I (Jul 2, 2008) Brief review of Part I 1 Finite fields Definitions Examples The structure of finite fields Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 3 / 18 Encoding methods Hamming distance as a metric Brief review of Part I (Jul 2, 2008) Brief review of Part I 1 Finite fields Definitions Examples The structure of finite fields 2 Classical codes over finite fields Linear codes: basic properties Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 3 / 18 Hamming distance as a metric Brief review of Part I (Jul 2, 2008) Brief review of Part I 1 Finite fields Definitions Examples The structure of finite fields 2 Classical codes over finite fields Linear codes: basic properties Encoding methods Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 3 / 18 Brief review of Part I (Jul 2, 2008) Brief review of Part I 1 Finite fields Definitions Examples The structure of finite fields 2 Classical codes over finite fields Linear codes: basic properties Encoding methods Hamming distance as a metric Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 3 / 18 We can define the sum and product of two polynomials using the usual rules of addition and multiplication over F. Example: let f (x) = x2 + 2x + 1 and g(x) = 2x + 1 be two polynomials over F3. Then f (x) + g(x) = x2 + x + 2 and f (x)g(x) = 2x3 + 2x2 + x + 1: The polynomials over a finite field F form an integral domain and are denoted by F[x]. Construction of finite fields Polynomials over finite fields Polynomials over finite fields Let F be a finite field. A polynomial over F is an expression of the form n X i n f (x) = ai x = a0 + a1x + ··· + anx i=1 where n = deg(f ) is a nonnegative integer called the degree of f (x), ai 2 F for all 0 6 i 6 n and x is a symbol not belonging to F, called an indeterminate over F. Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 4 / 18 Example: let f (x) = x2 + 2x + 1 and g(x) = 2x + 1 be two polynomials over F3. Then f (x) + g(x) = x2 + x + 2 and f (x)g(x) = 2x3 + 2x2 + x + 1: The polynomials over a finite field F form an integral domain and are denoted by F[x]. Construction of finite fields Polynomials over finite fields Polynomials over finite fields Let F be a finite field. A polynomial over F is an expression of the form n X i n f (x) = ai x = a0 + a1x + ··· + anx i=1 where n = deg(f ) is a nonnegative integer called the degree of f (x), ai 2 F for all 0 6 i 6 n and x is a symbol not belonging to F, called an indeterminate over F. We can define the sum and product of two polynomials using the usual rules of addition and multiplication over F. Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 4 / 18 The polynomials over a finite field F form an integral domain and are denoted by F[x]. Construction of finite fields Polynomials over finite fields Polynomials over finite fields Let F be a finite field. A polynomial over F is an expression of the form n X i n f (x) = ai x = a0 + a1x + ··· + anx i=1 where n = deg(f ) is a nonnegative integer called the degree of f (x), ai 2 F for all 0 6 i 6 n and x is a symbol not belonging to F, called an indeterminate over F. We can define the sum and product of two polynomials using the usual rules of addition and multiplication over F. Example: let f (x) = x2 + 2x + 1 and g(x) = 2x + 1 be two polynomials over F3. Then f (x) + g(x) = x2 + x + 2 and f (x)g(x) = 2x3 + 2x2 + x + 1: Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 4 / 18 Construction of finite fields Polynomials over finite fields Polynomials over finite fields Let F be a finite field. A polynomial over F is an expression of the form n X i n f (x) = ai x = a0 + a1x + ··· + anx i=1 where n = deg(f ) is a nonnegative integer called the degree of f (x), ai 2 F for all 0 6 i 6 n and x is a symbol not belonging to F, called an indeterminate over F. We can define the sum and product of two polynomials using the usual rules of addition and multiplication over F. Example: let f (x) = x2 + 2x + 1 and g(x) = 2x + 1 be two polynomials over F3. Then f (x) + g(x) = x2 + x + 2 and f (x)g(x) = 2x3 + 2x2 + x + 1: The polynomials over a finite field F form an integral domain and are denoted by F[x]. Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 4 / 18 5 4 Example: Consider f (x) = 2x + x + 4x + 3 2 F5[x], 2 3 2 g(x) = 3x + 1 2 F5[x]. Then q(x) = 4x + 2x + 2x + 1 and r(x) = 2x + 2, with deg(r) <deg(g). A polynomial with the leading term an = 1 is called a monic polynomial. A polynomial of degree zero is called a constant polynomial. Construction of finite fields Polynomials over finite fields Theorem 1: Division Algorithm Let g 6= 0 be a polynomial in F[x]. Then for any f 2 F[x] there exist polynomials q; r 2 F[x] such that f = qg + r; where deg(r) < deg(g): Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 5 / 18 A polynomial with the leading term an = 1 is called a monic polynomial. A polynomial of degree zero is called a constant polynomial. Construction of finite fields Polynomials over finite fields Theorem 1: Division Algorithm Let g 6= 0 be a polynomial in F[x]. Then for any f 2 F[x] there exist polynomials q; r 2 F[x] such that f = qg + r; where deg(r) < deg(g): 5 4 Example: Consider f (x) = 2x + x + 4x + 3 2 F5[x], 2 3 2 g(x) = 3x + 1 2 F5[x]. Then q(x) = 4x + 2x + 2x + 1 and r(x) = 2x + 2, with deg(r) <deg(g). Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 5 / 18 Construction of finite fields Polynomials over finite fields Theorem 1: Division Algorithm Let g 6= 0 be a polynomial in F[x]. Then for any f 2 F[x] there exist polynomials q; r 2 F[x] such that f = qg + r; where deg(r) < deg(g): 5 4 Example: Consider f (x) = 2x + x + 4x + 3 2 F5[x], 2 3 2 g(x) = 3x + 1 2 F5[x]. Then q(x) = 4x + 2x + 2x + 1 and r(x) = 2x + 2, with deg(r) <deg(g). A polynomial with the leading term an = 1 is called a monic polynomial. A polynomial of degree zero is called a constant polynomial. Vlad Gheorghiu (CMU) Finite fields: An introduction. Part II. August 7, 2008 5 / 18 A polynomial p 2 F[x] is said to be irreducible over F (or prime in F[x]) if p has positive degree and p = bc with b; c 2 F[x] implies that either b or c is a constant polynomial. 2 Example: x − 2 2 Q[x] is irreduciblep p over the field Q of rational 2 numbers, but x − 2 = (x + 2)(x − 2) is reducible over the field R of real numbers. Construction of finite fields Polynomials over finite fields Theorem 2: Greatest Common Divisor Let f1; f2;:::; fn be polynomials in F[x] not all of which are 0. Then there exists a uniquely determined monic polynomial d 2 F[x] with the following properties: (i) d divides each fj , 1 6 j 6 n; (ii) any polynomial c 2 F[x] dividing each fj , 1 6 j 6 n, divides d. Moreover, d can be expressed in the form d = b1f1 + b2f2 + ··· + bnfn; with b1; b2;:::; bn 2 F[x]: Vlad Gheorghiu (CMU) Finite fields: An introduction.
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