given error probability. Coding gain of a coded communication system over an uncoded system with the same target error probability. Coding gain= [L (in dB) code rate R to achieve error-free information transmission.See figures3 and 4 V.Finite Fields(分组码的代数结构) 1.Binary arithmetic and field Consider the binary se Define two binary multiplication,on (0,1)as follows +01 ·01 001 000 110 101 These two operations are commonly called module-2 addition and multiplication respectively.They can be implemented with an XOR and an AND gate,respectively. The)together with module-2 addition and multiplication called a binary field, denoted by GF(2)or F2 2.Vector space over GF(2) A binary n-tuple is an ordered sequence,(a1.a2.),with a,EGF(2). There are 2"distinct binary n-tuples Define an addition operation for any two binary n-tuples as follows: (a,a2,an)+(6,b2,bn)=(a1+h,a2+b2,an+bn) where is carried out in module-2 addition Define a scalar multiplication between and elementcin GF(2)and a binary n-tup (a1,2,.,an）as follows: c.(aa.)=(c.ac.azca) where ca,is carried out in module-2 multiplicatthe so set of all les.The set V"together with the addition oPaNe P3购oerldnu3s柳pE,Asn-Mu00Ne0yP3up space over GF(2).The elements in V are called vectors. -Note that V"contains the all-zero n-tuple (0,0.,0)and (a,42,a）+(a,42,a)=(0,0,0) 11 11 given error probability. „ Coding gain of a coded communication system over an uncoded system with the same modulation is defined the reduction, expressed in dB, in the required Eb/N0 to achieve a target error probability. Coding gain = 0 0 b b uncoded coded E E N N ⎡⎤ ⎡⎤ − ⎢⎥ ⎢⎥ ⎣⎦ ⎣⎦ (in dB) „ Shannon limit: A theoretical limit on the minimum SNR required for coded system with code rate Rc to achieve error-free information transmission. See figures 3 and 4. V. Finite Fields (分组码的代数结构) 1. Binary arithmetic and field „ Consider the binary set {0,1}, Define two binary operations, called addition ‘+’ and multiplication ‘·’, on {0,1} as follows + 0 1 0 0 1 1 1 0 „ These two operations are commonly called module-2 addition and multiplication respectively. They can be implemented with an XOR and an AND gate, respectively. „ The set {0, 1} together with module-2 addition and multiplication is called a binary field, denoted by GF(2) or F2 . 2. Vector space over GF(2) „ A binary n-tuple is an ordered sequence, (a1,a2,.,an), with GF(2) i a ∈ . - There are 2n distinct binary n-tuples. - Define an addition operation for any two binary n-tuples as follows: 1 2 12 1 1 2 2 ( , ,., ) ( , ,., ) ( , ,., ) n n nn aa a bb b a ba b a b + =+ + + where ai+bi is carried out in module-2 addition. - Define a scalar multiplication between and element c in GF(2) and a binary n-tuple (a1,a2,.,an) as follows: 12 1 2 ( , ,., ) ( , ,., ) n n c a a a ca ca ca ⋅ =⋅ ⋅ ⋅ where c⋅ai is carried out in module-2 multiplication. „ Let Vn denote the set of all 2n binary n-tuples. The set Vn together with the addition defined for any two binary n-tuples in Vn and the scalar multiplication is called a vector space over GF(2). The elements in Vn are called vectors. - Note that Vn contains the all-zero n-tuple (0,0,.,0) and 12 12 ( , ,., ) ( , ,., ) (0,0,.,0) n n aa a aa a + = · 0 1 0 0 0 1 0 1