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gb 2 1 X ; Y ! 1, or De2 X ; Y ! 2le2 1 if e1 > e2 g, or 2e fDe1 X ; Y ! t l e1 1 ^ gb 2 1 X ; Y ! 1 or ge2 X ; Y 0 if e1 > e2 g 1 2e gb2 e e
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De2 X ; Y ! 2le1 1 ^ gb3 1 X ; Y ! l 1 , e le2 =b EL-me3 =b ED code g12;3 X ; Y 0, De X ; Y ! l m e 1, 2;3 2;3 gb2 1 X ; Y ! m 1 if e2 e e3 , or De2 X ; Y ! 2le2 1 ^ gb2 e3 1 X ; Y ! 1 if e2 > e3 e
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Source: [KITA97]. 1997 IEICE Japan. & ei if ei ! ej ; Note: : ei;j max ei ; ej ej if ei < ej :
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CODES FOR ERROR LOCATION: ERROR LOCATING CODES
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TABLE 9.4 Code Conditions of Existing Codes Codes SEC code DED code SEC-DED code tb EC-mb ED code SEC-SbED code SEC-DED-SbED code Code conditions D1 X ; Y ! 3 D1 X ; Y ! 3 D1 X ; Y ! 4 Db X ; Y ! t m b 1 gb X ; Y ! 2, or Db X ; Y ! 2b 1 2 gb X ; Y 2 ! 2, 1 Db X ; Y ! 2b 1 ^ gb X ; Y ! 1 , or 3 Db X ; Y ! 2b 1 ^ D1 X ; Y ! 4 SbEC-DED code Db X ; Y ! 3b 1, or gb X ; Y ! 2 SEC-Se=b EL code 2 ^ Db X ; Y ! 2b 1 1 1 1 e gb X ; Y ! 1, 2e 1 gb X ; Y ! 1 ^ ge X ; Y 0 , e 2 1 gb X ; Y ! 2 ^ ge X ; Y 0 , or 2 1 De X ; Y ! 2e 1 gb X ; Y ! 1, 2e 1 SEC-DED-Se=b EL code De X ; Y ! 3e 1, ge X ; Y 0 ^ gb X ; Y ! 1 , or 1 e 2 fDe X ; Y ! 2e 1 ^ Sb=p b EL code gb X ; Y ! e 2 1, ge X ; Y 0, or gb X ; Y ! 2 g 1 2 gp X ; Y ! 1, g1 X ; Y 0, or 3 1 Dp X ; Y ! 2p 1 SbEC-Si b=p b ED code gp X ; Y ! 1, gp X ; Y ! 2, or i 2 i 1 Di X ; Y ! 2i 1
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Source: [KITA97]. 1997 IEICE Japan.
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Note: The means that the code conditions of the DED code, for example, are induced by taking logical ANDing the conditions of the me3 =b ED codes given by substituting two cases of m 1; e3 2 and m 2; e3 1 .
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Table 9.4 shows the conditions of the existing codes induced by substituting appropriate values to the parameters of t; l; m; e1 ; e2 , and e3 of the codes in Table 9.3. 9.6.3 Relation between Error Locating Codes and Error Correcting / Detecting Codes The necessary and suf cient conditions of the error locating codes include the condition of ge X; Y 0 in addition to the conditions of the error correcting codes. The necessity of 1 including this condition makes the number of codewords of error locating codes greater than that of the error correcting codes. The same is true for the relation between the le2 =b EL-me3 =b ED codes and the te1 =b EC-me3 =b ED codes. As the value of e becomes equal to b, the number of codewords that satisfy ge X; Y 0 becomes small, and therefore the error locating codes has to be equal to the 1 error correcting codes. If e b, the following theorem holds.
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