IV.Conventional Coding 1.Types of codes [block codes-linear codes,cyclic codes convolutional codes classfication based on structure [random-error-correcting codes burst-error-correcting codes (Binary codes Nonbinary codes error-correction codes error-detection codes 2.Error correcting capacity The error correcting capacity of a code Cdepends on its distance structure. The Hamming distance between two codewords,x and y,in a code,denoted by d(x,y),is defined as the number of places in which they differ. 4L)=立4Gdk,)=fx≠y 10,ifx,=y. ord(y){i:x≠yl For example,dy(010,111)=2. d(30102,21103)=3 Hamming distance satisfies the axioms for a distance metric: I)dm(x,y)≥0，with equality iffx=y 2)dn(,y)=dn(y,x)(对称性) 3)du(x.y)sdu(x,z)+du(z.y) The minimum Hamming distance of a code C is defined as dmin{dn(k,y)lx,y∈C,x≠y} For a convolutional code.this minimum Hamming distance is usually called the minimum free distance,denoted byd An(n,k)block code with minimum Hamming distance d is capable of correcting 7 IV. Conventional Coding 1. Types of codes block codes - linear codes, cyclic codes classfication based on structure convolutional codes ⎧ ⎫ ⎨ ⎬ ⎩ ⎭ random-error-correcting codes burst-error-correcting codes ⎧ ⎨ ⎩ Binary codes Nonbinary codes ⎧ ⎨ ⎩ error-correction codes error-detection codes ⎧ ⎨ ⎩ 2. Error correcting capacity „ The error correcting capacity of a code C depends on its distance structure. „ The Hamming distance between two codewords, x and y, in a code, denoted by H d (, ) x y , is defined as the number of places in which they differ. H H 1 H 1, if ( , ) ( , ), ( , ) 0, if or ( , ) |{ : }| n i i Hii ii i i i i i x y d d xy d xy x y d ix y = ⎧ ≠ = = ⎨ ⎩ = = ≠ x y ∑ x y For example, (010,111) 2, (30102,21103) 3 H H d d = = „ Hamming distance satisfies the axioms for a distance metric: 1) ( , ) 0, with equality iff H d x y ≥ = x y 2) ( , ) ( , ) ( ) H H d d xy yx = 对称性 3) ( , ) ( , ) ( , ) H HH d dd x y ≤ + xz z y „ The minimum Hamming distance of a code C is defined as d d min  min ( , ) | , , { H xy xy x y ∈C ≠ } „ For a convolutional code, this minimum Hamming distance is usually called the minimum free distance, denoted by free d . „ An (n, k) block code with minimum Hamming distance min d is capable of correcting