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Theorem 13.8 The null space of H given in Eq. (13.6) is a single adjacent-symboltransposition error correcting code over GF q . Proof Let X x0 x1 . . . xi 2 xi 1 xi xi 1 . . . xn 1 be a transmitted codeword, and let X 0 x0 x1 . . . xi 2 xi xi 1 xi 1 . . . xn 1 be a received word having single adjacent-symbol-transposition error in xi 1 and xi . Syndrome S for X 0 is given as follows: S X 0 HT xi bi 1 xi 1 bi xi 1 bi 1 xi bi xi 1 xi bi bi 1 : Hence the adjacent-symbol-transposition errors can be corrected if the following condition is satis ed: bi bi 1 6 a bj bj 1 for all i; j 2 f1; 2; ; n 1g; and for all a 2 GF q f0g; where i < j. Suppose that there exist i; j; and a satisfying bi bi 1 a b j b j 1 ; then the following equations hold: bi 1 b b0 ab j 1 b b0 ; bi 1 ab j 1 ; bi 1 q 1 ab j 1 q 1 ; b i 1 q 1 b j 1 q 1 : , a q 1 1 The equations above imply i j because b is a primitive element in GF qr and 0 i 1 q 1 < j 1 q 1 < qr 1. This contradicts the hypothesis of i < j, and hence Eq. (13.7) holds. Q.E.D. By Theorems 13.7 and 13.8, the null space of H can be used as either a single-symbol error correcting code or a single adjacent-symbol-transposition error correcting code. Note that the code de ned by H cannot distinguish between single-symbol errors and single adjacent-symbol-transposition errors. 13.3.2 Code Design 13:7
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The codes are de ned over the set of M-ary symbols A fa0 ; a1 ; ; aM 1 g. For given asymmetric error probabilities p aj jai and the threshold error probability e, error directionality graph G is de ned in De nition 13.3. Mapping f is de ned as a vertex coloring function for the error directionality graph G, shown below.
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De nition 13.11 For the set of M-ary symbols A and Galois eld GF q , f is de ned as a mapping from A to GF q satisfying the following condition: ai ! aj 2 E ! f ai 6 f aj for all ai ; aj 2 A; ai 6 aj ; where E is the set of edges in the error directionality graph G V; E . &
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De nition 13.12 For the set of M-ary symbols A and the set of integers ZM f0; 1; . . . ; & M 1g, g is de ned as an arbitrary bijective mapping from A to ZM . The following theorem gives a new class of nonsystematic M-ary codes capable of correcting single deletion / insertion errors and single adjacent-symbol-transposition errors as well as correcting single asymmetric errors [KANE04c]. Theorem 13.9 Let C be a set of codewords u u0 u1 . . . un 1 satisfying the following three conditions: P  n 1 1. i 0 g ui mod M 0, P  n 2 i 1 g ui mod n 0, 2. i 0 3. f u0 ; f u1 ; . . . ; f un 1 HT 0, where ui 2 A fa0 ; a1 ; . . . ; aM 1 g for all i 2 f0; 1; . . . ; n 1g, g u0 g u1 . . . g un 2 is the associate vector for g u0 g u1 . . . g un 1 , H is the parity-check matrix given by Eq. (13.6), and 0 is a zero vector. The code C is a nonsystematic M-ary asymmetric error correcting code with deletion / insertion / adjacent-symbol-transposition error correction capabilities. In other words, a received word v v0 v1 . . . vn0 1 is correctly decoded if v has no error or has one of the following errors: a. Single asymmetric errors, b. Single deletion / insertion errors, c. Single adjacent-symbol-transposition errors existing in vi ; vi 1 , where f vi 6 f vi 1 . Here a single asymmetric error is a single-symbol error indicated by an edge ui ! uj 2 E, that is, an error in which symbol ui is changed to another symbol vi with probability p vi jui > e . Proof Let u u0 u1 . . . un 1 be a codeword having length n, and let v v0 v1 . . . vn0 1 be a received word having length n0 . Syndromes S1 and S2 for v is determined as follows: ! n0 1 X g vi mod M; S1
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