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(b) Error directionality graph GC
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Figure 13.2 Asymmetric errors in numeric keypad. Source: [KANE04a]. 2004 IEEE.
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This error set includes all pairs of symbols corresponding to adjacent keys in the numeric keypad. Figure 13.2(b) shows the error directionality graph GC based on C . De nition 13.4 Let u u0 ; u1 ; ; uN 1 be a codeword of code C over the set of M-ary symbols A fa0 ; a1 ; ; aM 1 g, i.e., ui 2 A 0 i N 1 . If the code C can correct every single-symbol error ui ! u0i 2 , then C is an M-ary single asymmetric symbol error correcting code. & Bound for Single -Asymmetric Symbol Error Correcting Codes Here we observe the systematic M-ary codes that are capable of correcting single -asymmetric symbol errors that occur in the check part as well as those in the information part. However, in order to derive the upper bound on the information symbol length of the codes, Lemma 13.1 deals with the codes capable of correcting single -asymmetric symbol errors only in the information part. De ne the following functions:     d ai f aj ! ai j aj ! ai 2 E g;   G
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where jXj denotes the cardinality of X. In other words, d ai is the indegree of ai , and G is the maximum indegree of vertices in G. Lemma 13.1 A systematic code that corrects single -asymmetric symbol errors in the information part exists only if " r # M 1 k ; G where bxc shows the maximum integer less than or equal to x, k is the information-symbol length, r the check-symbol length, and G the error directionality graph based on . Proof Let u1 and u2 be any two distinct codewords expressed by a a a a u1 e; ; e; a0 ; e; ; e; p1;0 ; ; p1;r 1 ; | {z } | {z } | {z }
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where e; a0 ; a00 ; p1; j ; p2; j 2 V A 0 a d e G ; 0 a a0 ! e 2 ; a s; t k 1; j r 1 ;
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a and a00 ! e 2 . If u1 and u2 have identical check parts (i.e., p1; j p2; j pj for 0 j r 1), then both u1 and u2 may be changed into the following identical word by a single -asymmetric symbol error in the information part: u0 e; e; ; e; p0 ; ; pr 1 : a a a | {z } | {z }
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In this case errors in u1 and u2 cannot be corrected. Therefore all codewords with the following information part should not have an identical check part: e a
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k 1. So the inequality Mr
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is satis ed because there are k G 1 codewords with this property. Subsequently the inequality in Lemma 13.1 holds. Q.E.D. This lemma indicates that any code capable of correcting single -asymmetric symbol errors in the information part has at most b M r 1 = G c information symbols. Since M-ary single -asymmetric symbol error correcting codes include this error correction capability, the information symbol length of these codes never exceeds b M r 1 = G c. The following theorem is obvious. Theorem 13.1 exists only if A systematic M-ary single -asymmetric symbol error correcting code k " r # M 1 : G
Rings We de ne a new class of rings on which the M-ary single -asymmetric symbol error correcting codes are designed. De nition 13.5 Let c be a positive integer, and also qi be a prime number or a power of a prime number, where 1 i c. A set R q1 ; q2 ; ; qc is de ned by R q1 ; q2 ; ; qc fhx1 ; x2 ; ; xc i j xi 2 GF qi ; 1 i cg:
Let x hx1 ; x2 ; ; xc i and y hy1 ; y2 ; ; yc i be elements in R q1 ; q2 ; ; qc . Addition and multiplication in R q1 ; q2 ; ; qc are de ned as follows: x y hx1 ; x2 ; ; xc i hy1 ; y2 ; ; yc i h x1 1 y1 ; x2 2 y2 ; ; xc c yc i; x y hx1 ; x2 ; ; xc i hy1 ; y2 ; ; yc i h x1 1 y1 ; x2 2 y2 ; ; xc c yc i; where i and i 1 respectively. Theorem 13.2 i c are additive and multiplicative operators in GF qi , &