y+=000.(0001.(0010.(01,.(0100.(0101.(0110,.(01 11000,(1001),(1010),101),(1100),(1101),1110),1110) Asubset Sof V is called a subspace of V"if 1)the all-zero vector is in S. 2)the sum of two vectors in S is also a vector in S: For example:(0000),(0101),(1010),(1111);forms a subspace ofv A linear combination ofk vectors,V,V,.V,in V is a vector of the form V=cV+cV2+.+caV where c,EGF(2)and is called the coefficient of V. The subspace formed by the 2 linear combinations of linearly independent vectors Vi.V3.V.in Va is called a k-dimensional subspace of V A binary polynomial is a polynomial with coefficients from the binary field. For example,1+x2,1+x+x -A binary polynomial p(x)of degree m is said to be irreducible if it is not divisible by nial of egree less than m and greater than zero.For example.1+x+x -An irreducible polynomial p(x)of degree m is said to be primirive if the smallest positive integer n for which p(x)divides+1 is n=2-1.For example,p(x)=1+x+x.(it divides x+) -For any positive integer m,there exists a primitive polynomial of degree m.( Groups:Agroup is an alebraic structure (G)consisting of set Gand an operation satisfying the following axioms: 1)Closure:For any a,beG,the element atb is in G; 2)Associative law y For a,b,cEG,a*(b*c)=(a*b)*c 3)Identity element:There is an element eG for whichea=ae=a for all aG: 4)Inverse:For every aeG there exists a unique element aG such that a*a= al#a=e. Agroup is called a commutative group or Abelian group ifa*b=b*a for all a,beG Examples:-整数，有理数，实数，with addition. Integers with module-m additior Fields:A field F is a set that has two operations defined on it:Addition and multiplication,such that the following axioms are satisfied: I)The set is an Abelian group under addition;(单位元称为o') 13 13 4 (0000),(0001),(0010),(0011),(0100),(0101),(0110),(0111) (1000), (1001), (1010), (1011), (1100), (1101), (1110), (1110) V ⎧ ⎫ = ⎨ ⎬ ⎩ ⎭ „ A subset S of Vn is called a subspace of Vn if 1) the all-zero vector is in S; 2) the sum of two vectors in S is also a vector in S； For example: S={(0000),(0101),(1010),(1111)} forms a subspace of V4 . „ A linear combination of k vectors, 1 2 , ,., VV Vk in Vn is a vector of the form 11 2 2 k k VV V V = cc c + ++ " where GF(2) i c ∈ and is called the coefficient of Vi. „ The subspace formed by the 2k linear combinations of k linearly independent vectors 1 2 , ,., VV Vk in Vn is called a k-dimensional subspace of Vn . „ A binary polynomial is a polynomial with coefficients from the binary field. For example, 1+x 2 , 1+x+x 3 . - A binary polynomial p(x) of degree m is said to be irreducible if it is not divisible by any binary polynomial of degree less than m and greater than zero. For example, 1+x+x2 , 1+x+x3 . - An irreducible polynomial p(x) of degree m is said to be primitive if the smallest positive integer n for which p(x) divides x n +1 is n=2m-1. For example, p(x)=1+x+x4 . (it divides x15+1) - For any positive integer m, there exists a primitive polynomial of degree m. (可查表) 4. Galois fields „ Groups: A group is an algebraic structure (G, *) consisting of a set G and an operation * satisfying the following axioms: 1) Closure: For any a, b∈G, the element a*b is in G; 2) Associative law: For any a, b, c∈G, a*(b*c)=(a*b)*c; 3) Identity element: There is an element e∈G for which e*a=a*e=a for all a∈G; 4) Inverse: For every a∈G, there exists a unique element a-1 ∈G, such that a*a-1 = a -1*a = e. „ A group is called a commutative group or Abelian group if a*b = b*a for all a, b∈G. Examples: - 整数，有理数，实数，with addition; - Integers with module-m addition. „ Fields: A field F is a set that has two operations defined on it : Addition and multiplication, such that the following axioms are satisfied: 1) The set is an Abelian group under addition; (单位元称为‘0’)