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P i; j where denotes summation of matrices over GF 2 excluding r i; j check bits in UARC . The arithmetic residue-check part UARC is determined by the following procedure: Step 1. Arithmetic residue checks ARCV and ARCH are written as
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!! w Ui; j !! mod MH ; mod MV ;
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where MV 2mlm ln 1; MH n 1 lm ln 1; U0;0 UPC , and w Ui; j is the num i; j ber of 1 s in Ui; j excluding that in UARC .
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Step 2. Let ARCV 2 and ARCH 2 be binary representations of ARCV and ARCH , respectively. These binary vectors are concatenated in order to generate a vector ARCV 2 ; ARCH 2 having length K dlog2 MV e dlog2 MH e bits. Step 3. Let r ln be the minimum integer satisfying the inequality r mn 1 ! R K; r K 13:10
Also let 0 0 0 be a zero vector having length kr K bits, where k dK=re. The binary vector 0; ARCV 2 ; ARCH 2 is equally divided into k binary blocks each having length r bits, |{z} ; ARCV 2 ; ARCH 2 Dk 1 ; Dk 2 ; ; D0 ; 0 | {z } | {z }
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where Di , 0 i k 1, is a binary block having length r bits. Step 4. The binary vector Dk 1 ; Dk 2 ; ; D0 is encoded by 1D-Ur BEC code as shown previously, and the codeword obtained is V Dk 1 Dk 2 D0 PC1D f ARC1D : Step 5. The leftmost kr K bits in V are removed, and the remaining part V 0 of V is i; j divided into mn 1 binary vectors, each of which corresponds to a vector UARC where i; j 2 . The resulting arithmetic residue-check part is UARC V 0 UARC ; UARC ; ; UARC ; UARC ; UARC ; ; UARC where length r i; j of UARC satis es
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Figure 13.21 illustrates this residue-check procedure. If the condition given by the inequality (13.10) is not satis ed for any r ln, the 2D-Ulm ln EC code cannot be designed, and then the conventional two-dimensional clustered error correcting codes [BREI98] are used.
(i+1) w(Ui,j)
mod M V
mod M H
[ARC V]2 0...0
Binary representations Concatenated
[ARC H]2
( [ARC V]2, [ARC H]2 ) K bits ...
Divided into binary blocks
D k-1 D k-2
V = D k-1 D k-2
Encoded by 1D-U BEC code ...
PC1D f(ARC 1D)
(m-1, U ARC n-1)
(0,1) (0,2) U ARC = U ARC U ARC or -1 bits
Figure 13.21 Generation of UARC. Source: [KANE03]. 2003 IEEE.
U 0,0
... U =
... U n-1 0, ... : Received word
U -1,0 ... U -1,n-1 m m
Decode U ARC of U by 1D-U BEC code and determine ARC V and ARC H Syndrome calculation:
Each bit in U ARC =0
SP = U j , SV = (i+1) , (U j ) -ARCV mod M V SH = j w(U j ) -ARC H mod M H w i, i, i,
i=0 j=0 i=0 j=0 j=0 i=0
m-1 n-1
S V = M V -SV SH = (M H -SH) mod M H
F 1-error
_ _ 1 < SV < mlml n
no error
T 0-error SV SH -1 t= A=SV mod w(SP) B=SH mod w(SP ) w (SP ) w (SP)
Divide SP into 4 submatrices: SP =
E1,1 E0,1
E1,0 E0,0
where w (E1,0)+ w (E1,1) =A w (E0,1)+w (E1,1)=B
Error correction in
U t ,U t+1 , U t , and U t+1 : s, s, s+1, s+1,
... U t s, ...
U t+1 ... s,
... O + ... O
... U t U t+1 ... s+1, s+1,
E0,0 E0,1 E1,0 E1,1
... ...
Figure 13.22 Decoding procedure. Source: [KANE03]. 2003 IEEE.
Theorem 13.10 U lm ln EC code.
Codeword U obtained by the above procedure is a codeword of 2D-
(3) Decoding Procedure Ulm ln EC code. Example 13.8
Figure 13.22 illustrates the decoding procedure of the 2D-
Encoding The objective is to design the 2D-U3 3 EC code with code parameters m 3 and n 4. We have MV 55; MH 28; K dlog2 MV e dlog2 MH e 11; r 2; and R 11; 2 K 9 11 20. From the information part shown in Figure 13.23 (a), the parity-check part is determined as 0 1 40 1 0 1 2 3 1 0 5: 1
Then ARCV and ARCH are calculated as follows: ARCV 1 17 2 12 3 17 mod 55 37; ARCH 0 14 1 8 2 14 3 10 mod 28 10: