CODES FOR HIGH-SPEED MEMORIES III: BIT / BYTE ERROR CONTROL CODES in .NET

Deploy qr bidimensional barcode in .NET CODES FOR HIGH-SPEED MEMORIES III: BIT / BYTE ERROR CONTROL CODES
CODES FOR HIGH-SPEED MEMORIES III: BIT / BYTE ERROR CONTROL CODES
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Example 6.10 For b 4 and A 2, the sub eld GF 22 can be used to factor GF 24 into l 5 multiplicative cosets as follows: Q0 Q1 Q2 Q3 Q4 f0; f0; f0; f0; f0; T5 ; T6 ; T7 ; T8 ; T9 ; T10 ; T11 ; T12 ; T13 ; T14 ; T15 Ig GF 22 sub eld of GF 24 Tg T Q0 T2 g T2 Q0 T3 g T3 Q0 T4 g T4 Q0 :
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Lemma 6.4 Every element in a multiplicative coset of the GF 2A sub eld of GF 2b is closed under addition in GF 2b . This lemma can be easily proved, and therefore the proof is omitted. The lemma leads to the next theorem [FUJI91a, 91b]. Theorem 6.16 With elements in a multiplicative coset of the GF 2A sub eld of GF 2b , A namely Qd f0; T l d ; T 2l d ; . . . ; T 2 1 l d g, where d 0, 1, . . ., l 1, the following H matrix shows the SbEC-DED code with maximum code length in bits N b 2A : 2 6 H 6 4 Ib j0R b jb Ib  l d  T 
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 il d  T R b b
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where j0R b jb and jT il d jb are R b b binary slimmed elements of the original ones in R b Qd , and I b is an identity element in GF 2b . It is apparent that the SbEC-DED code having multiple rows in H can also be designed by using the elements of a multiplicative coset as shown in Theorem 6.15. Design by Using Elements of Additive Coset The nite eld GF 2b can be factored into 2b A additive cosets by the sub eld GF 2A . Every element of the GF 2b eld is in one and only one coset of a sub eld GF 2A . Example 6.11 For b 4 and A 2, there exist four additive cosets of the GF 22 sub eld of GF 24 as shown below. In this case the companion matrix T is determined by the primitive polynomial g x x4 x 1: P0 f0; T5 ; T10 ; T15 Ig GF 22 subfield of GF 24 P1 fT; T2 ; T4 ; T8 g T P0 P2 fT3 ; T11 ; T12 ; T14 g T3 P0 P3 fT6 ; T7 ; T9 ; T13 g T6 P0 :
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SINGLE-BYTE ERROR CORRECTING AND DOUBLE-BIT ERROR DETECTING (SbEC-DED) CODES
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Lemma 6.5 The addition with two elements in an additive coset results in an element of the sub eld GF 2A . This lemma can be easily proved by using the property of the additive coset. So we have the following SbEC-DED code [HAMA91, FUJI93]. Theorem 6.17 For using elements in an additive coset of the GF 2A sub eld of GF 2b , the following H matrix shows the SbEC-DED code with maximum code length in bits N b 2A : " H2 Ib T p0 Ib T p1 Ib T p2 Ib T pi Ib T p2A 1 # ;
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where I b is an identity element in GF 2b and T pi s, i 0, 1, 2, , 2A 1, are elements in an additive coset. From this theorem, it is apparent that the H matrix with multiple rows that satis es linear independence between any pair of columns gives the SbEC-DED code with using elements in an additive coset. That is, the H matrix has the following structure: all I elements are included in the rst row and the remaining part of H with r 1 rows has different column vectors from the elements of an additive coset, as shown below: 2 I 6 6 6 6 Hr 6 6 6 4 I I Ti 1 Ti2 . . . ir 1 T I Tj1 Tj2 . . . jr 1 T 3 I 7 7 7 7 7: 7 7 5
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In this case, Ti1 , Ti2 , . . ., Tir 1 , Tj1 , Tj2 , . . . , Tjr 1 , are elements included in the additive coset of GF 2A and I is an identity element in GF 2b . The maximum number of column vectors, meaning the maximum code length n (bytes), can be obtained as n 2A r 1 2A r 1 : 6:10
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By using the Hr with r rows above de ned, we obtain the new SbEC-DED code with an extended code length in the next theorem. Theorem 6.18 The following H matrix shows the SbEC-DED code with a maximum code length in bits N b 2A r 1 2A r t 1 , where 1 t r 2, elements in H
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