The direct method algorithm and the LTD Algorithm

CHAPTER 3 CONSTRUCTION OF LOOKUP TABLES AND BINARY-SEARCH

4.2 The direct method algorithm and the LTD Algorithm

1 ( 1



S v could be found in the table and we obtain the error patterns e , According to the _m

LTD algorithm, the receive vector could be corrected as rm^' rm em ep r_m, where e p

is set of binary 2

1

n -tuples in which only the p-th coordinate contains a nonzero value at

p-th position.

The flowchart of the new decoding algorithm is depicted in Figure II. The LTD algorithm, shown in Figure II, has also been programmed in C language.

4.2 The direct method algorithm and the LTD Algorithm

As long as the table is constructed, the direct method algorithm is thus applied to decode the QR codes which can be summarized by the following ten steps:

The direct method algorithm 1) Initialized by letting p k1.

2) Compute the syndromes S₁ for a received vector r.

3) If S₁=0 go to stop.

4) If syndrome is in T, go to step 9.

5) Check S₁+^p value in T. If syndrome is in T, go to step 8.

6) Compute p p1.

7) If p>n-1, go to stop. Otherwise return to step 5.

8) If the index corresponding to the syndrome in the lookup table A, then the error pattern

em is obtained by index to find in lookup table B. Addition e^'_m by one bit at p-th position, obtaining error pattern e_m (i.e., em em^' ep where ep is a binary k-tuples in which contains only a nonzero value at p-th position), go to step 10.

9) Look for the index corresponding to the syndrome in the lookup table A, then error pattern e_m is obtained by index to be found in the lookup table B.

10) Subtract the error pattern e_m from received vector r_m, obtaining the message block c_m, go to stop.

The flowchart of the direct method algorithm is shown in Fig. 1.

To decode all the (23, 12, 7), (41, 21, 9) and (47, 24, 11) QR codes, we construct sorted syndrome values in the memory, namely set T. The set T is designated as ^T ^



_v ^S¹⁽^v⁾ ^{. This}

algorithm is summarized by the following ten steps:

The LTD algorithm

1) Initialized by letting p0.

2) Compute the syndromes S₁ for a received vector r.

3) If S₁0 go to stop.

4) If syndrome is in T, go to step 9.

5) Check S₁^p value in T. If syndrome is in T, go to step 8.

6) Compute 1p  p .

7) If pnt, go to stop. Otherwise return to step 5.

8) If 1p k ,look for the index corresponding to the syndrome in the lookup table A, then the error pattern e^'_m is obtained by index to find in lookup table B. Addition e^'_m

by one bit at p-th position, obtaining error pattern e_m (i.e., em em^' ep where ep is a binary k-tuples in which the p-th coordinate contains a nonzero value only at p-th position), go to step 10.

9) Look for the index corresponding to the syndrome in the lookup table A, then error pattern e_m is obtained by index to be found in the lookup table B.

10) Subtract the error pattern e_m from received vector r_m, obtaining the message block c_m,

go to stop.

The above new decoding schemes have been verified exhaustively for v errors, where t

v



0 , by a software simulation. The new decoding schemes are suitable for both software and hardware realizations. However, in searching the syndrome of t errors case, we could not avoid that all 1 to t-1 error patterns occurred at remainder d(x). In this situation, the error would be found after executing the shift-search algorithm. Therefore, some computer simulation time would be increased. The probability of all errors appeared in the remainder of each QR code is listed in Table 4.1.

Table 4.1 The probabilities of all error patterns that are all appeared in the remainder for each QR code

Number of errors

(23, 12, 7) (41, 21, 9) (47, 24, 11)

1 11/23 20/41 23/47

2 55/253 190/820 253/1081

3 1140/10660 1771/16215

4 8855/178365

The computer simulation comparisons among errors of QR codes with different methods are shown in the Table 4.2. In computer simulation, we test 1,000,000 codewords for each error case. According to the simulation results, all codewords can be decoded within 4*10^-6 second. It is quite practical result that can be utilized in the SOC software.

Moreover, the memory requirement of the direct method algorithm is about ten times larger than of the LTD algorithm, but the CPU time of the direct method algorithm is half of the LTD algorithm. In other words, the direct method algorithm increases the memory requirement up to 0.32K bytes and 12Kbytes than the LTD algorithm. However, the CPU time of the direct method algorithm is half of the LTD algorithm except in decoding the t-th error.

Table 4.2 Comparisons among different methods for each QR code QR Code v errors Direct method LTD algorithm

1 0.27s 0.75s

2 0.42s 0.92s

(23, 12, 7)

3 0.58s

10K Bytes

1.34s

1.03K Bytes

1 0.79s 2.36s

2 0.95s 1.89s

3 1.04s 1.93s

(41, 21, 9)

4 1.10s

991K Bytes

2.92s

89.4K Bytes

1 _{0.516 s} _{3.765 s}

2 _{1.125 s} _{2.968 s}

3 _{1.515 s} _{1.984 s}

4 _{2.063 s} _{1.782 s}

(47, 24, 11)

5 _{2.094 s}

9640 K Bytes

3.884 s

1050 K Bytes

Chapter 5 Conclusions and Recommendations for Further Research

5.1 Conclusions

In this dissertation, we present the direct method algorithm and the LTD algorithm to decode the binary systematic (23, 12, 7), (41, 21, 9) and (47, 24, 11) QR codes, respectively.

Such algorithms determine directly the error locations by lookup tables without the operations of addition and multiplication over a finite field. According to Table 4.2, the memory requirement of the direct method algorithm is about ten times larger than the LTD algorithm, but the CPU time of the direct method algorithm is half of the LTD algorithm. In other words, the direct method algorithm sacrifices some memory requirement to exchange CPU time.

On the other hand, if we take (41, 21, 9), for example, to execute the algorithm in [15], it needs at least a memory requirement of up to 6Mbyte for software implementation of arithmetic operations in a Galois Field GF(2²⁰). The time of addition and multiplication operations is not included. It is too huge to be implemented in the SOC system. Therefore, the LTD algorithm mentioned in this paper in terms of both speed and memory requirements could be utilized in real time systems.

5.2 Performance Analysis

Fig. 2 compares the Bit-Error Rate (BER) of Binary Phase-Shift Keying (BPSK), (23, 12,

7), (41, 21, 9) and (47, 24, 11)QR Codes through AWGN channel. A (n, k, t) Binary QR code can correct up to t errors, where t is directly proportional to the number of parity bits.

Hence the performance of various binary QR codes are analyzed by varying n and k and their corresponding energy consumption is measured.

It is clear from Fig. 5.1 that (47, 24, 11) QR code, which can correct up to 5 errors has the lowest BER curve with the highest coding gain.

Bit-error rate versus Eb/No

1.00E-06 1.00E-05 1.00E-04 1.00E-03 1.00E-02 1.00E-01 1.00E+00

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Eb/No (dB)

Bit-error rate BPSK

QR23 QR41 QR47

Fig. 5.1 Performance analysis of BPSK and QR codes

5.3 Recommendations for Further Research

The above dissertation of the systematic binary QR codes suggests the following problems for further research:

1. The research accomplished here can be extended to other QR codes.

2. As long as the codeword length becomes longer, the limit of lookup table method would occur. For example, in the (87, 45, 17) QR code, the sum of all error patterns are:



i i . It is impossible to construct such huge table in hardware. Therefore, some new algorithms or ideas need to be considered.

3. The algorithms mentioned in this dissertation could be utilized in hardware design.

4. The research accomplished here can not applied to decode some systematic binary QR codes directly because the elements in Galois Field is fewer than the sum of all correctable error patterns. For example, in (31, 16, 7) QR code, the Galois Field in 2 which contains only ⁵ 2⁵131 elements. However, the sum of all

correctable error patterns are 4991 3

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Appendix A

A.1 Construct the Lookup Tables of the (23, 12, 7) QR Code

The (23, 12, 7) QR code could correct up to 3 errors. But in LTD algorithm, we only store t-1=2 error patterns in the lookup tables. The detail of construct tables is shown below.

Case 0) For no error occurred, the received codeword, the syndromes is S₁⁽⁰⁾ 0 Case 1) For 1 error, S1⁽¹⁾ {s|sⁱ for 11i23}.

Case 2) For 2 errors, S1⁽²⁾ {s|sⁱ ^j for 11i23 , i j 0}.

Let e(e_d,e_m)(e₀,e₁,,e_k_₂,e_k_₁,e_k,,e_n_₁), where e_d (e₀,e₁,e_k_₁) and )

, , (

e_m  e_k_₁ e_n_₁ be the error vector and ei , where 0 i22, be a set of binary 23-tuples in which only the ith coordinate contains a nonzero value. For example,

) 0 , , 0 , 1 , 0

1 ( 

e . With the above definitions, the pseudo makes up two tables as follows:

Step 1: Initial index0.

Step 2: Error pattern of 1 for i  11 until 22 do begin

De_tableA[index]record of  ; ⁱ eei;

De_tableB[index]e_m; Index  index + 1 ; end;

Step 3: Error patterns of 2 for i  11 until 22 do for j  i-1 until 0 do begin

De_tableA[index]record of ⁱ ^j; eei  ; ej

De_tableB[index]e_m; index  index + 1;

end;

Step 4: De_table[] is sorted according to the sorting pattern of the syndrome array of De_tableA[].

A.2 Construct the Lookup Tables of the (41, 21, 9) QR Code

The (41, 21, 9) QR code could correct up to 4 errors. But in LTD algorithm, we only store t-1=3 error patterns in the lookup tables. The detail of construct tables is shown below.

Case 0) For no error, S₁⁽⁰⁾ 0.

Case 1) For 1 error, S₁⁽¹⁾ {s|sⁱ for 20i41}.

Case 2) For 2 errors, S₁⁽²⁾ {s|sⁱ  ^j for 20i41, i j 0}

Case 3) For 3 errors, S₁⁽³⁾ {s|sⁱ ^j ^k for 20i41, i jk0}.

Let e(e_d,e_m) (e₀,e₁,,e_k_₂,e_k_₁,e_k,,e_n_₁) , where )

, , ,

( ₀ ₁ _₂

 _k

d e e  e

e and e_m (e_k_₁,,e_n_₁) be the error vector and e_i , where 40

0 i  , be a set of binary 41-tuples in which only the i-th coordinate contains a nonzero value. For example, e₁ (0,1,0,,0). With the above definitions, the pseudo code makes up two tables as follows:

Step 1: Initial index0.

Step 2: Error pattern of 1 for i  20 until 40 do begin

De_tableA[index]record of ⁱ; ei

e ;

De_tableB[index]e ; _m index  index +1;

end;

Step 3: Error patterns of 2 for i  20 until 40 do for j  i-1 until 0 do begin

De_tableA[index]record of ⁱ  ^j;

i e

e  ;

De_tableB[index]e_m; index  index +1;

end;

Step 4: Error patterns of 3 for i  20 until 40 do for j  i-1 until 1 do for k  j-1 until 0 do begin

De_tableA[index]record of ⁱ  ^j ^k;

k j

i e e

e   ;

De_tableB[index]e_m; index  index +1;

end;

Step 5: De_tableB[] is sorted according to the sorting pattern of the syndrome array of De_tableA[].

A.3 Construct the Lookup Tables of the (47, 24, 11) QR Code

The (47, 24, 11) QR code could correct up to 5 errors. But in LTD algorithm, we only store t-1=4 error patterns in the lookup tables. The detail of construct tables is shown below.

Case 0) For no error in the received codeword, the syndrome is S₁⁽⁰⁾ 0. Case 1) For 1 error, S₁⁽¹⁾ {s|sⁱ for 23i47}.

Case 2) For 2 errors, S₁⁽²⁾ {s|sⁱ^j for 23i47, i j0}.

Case 3) For 3 errors, S₁⁽³⁾ {s|sⁱ^j ^k for 23i47,i jk0}

Case 4) For 4 errors in the received codeword, the syndrome is

| 47-tuples in which only the ith coordinate contains a nonzero value. For example,

)

e . With the above definitions, the pseudo code makes up two tables as follows:

Step 1: Initial index0.

Step 2: Error patterns of 1 for i  23 until 46 do

Step 3: Error patterns of 2

for i  23 until 46 do

Step 4: Error patterns of 3 for i  23 until 46 do

Step 5: Error patterns of 4 for i  23 until 46 do indexindext+1;

end;

Step 6: De_tableB[] is sorted according to the sorting pattern of the syndrome array of De_tableA[].

Appendix B

The binary-search algorithm plays an important roll in both the direct method and the LTD algorithm. Therefore, we give the detail algorithm for all codes.

B.1 Binary-Search Algorithm for (23, 12, 7)

int Bin_Search(unsigned int syndrome, De_tableA[]){

int high = 210, low = -1, probe;

while (high-low >1){

probe=(low+high)>>1;

if (De_tablea[probe]>syndrome) high = probe;

else

low = probe;

}

if (low = =-1 ||De_tablea[low] !=syndrome) return -1;

else

return low;

}

B.2 Binary-Search Algorithm for (41, 21, 9)

int Bin_Search(unsigned int syndrome, unsigned int De_tableA[]) { int high =10171, low = -1, middle;

while (high - low > 1){

middle = (low + high) >> 1;

if (De_tableA[middle] > syndrome) high = middle;

else

low = middle;}

if (low == -1 || De_tableA[low] != syndrome) return -1;

else

return low;

}

B.3 Binary-Search Algorithm for (47, 24, 11 )

int Bin_Search(unsigned int syndrome, unsigned int De_tableA[]){

int high =184806, low = -1, middle;

while (high - low > 1){

middle = (low + high) >> 1;

if (De_tableA[middle] > syndrome) high = middle;

else

low = middle;

}

if (low == -1 || De_tableA[low] != syndrome) return -1;

else

return low;

}

Appendix C

C.1 Decoding Process of the LTD Algorithm in Decoding (23, 12, 7)

(1) Initialized by letting p=0.

(2) Compute the syndrome S₁ for a received vector r.

(3) If S₁0 go to stop.

(4) If syndrome is in T, go to step 9.

(5) Check S₁^p value in T. If syndrome is in T, go to step 8.

(6) Compute p p1.

(7) If p n3, go to stop. Otherwise return to step 5.

(8) If p k1, look for the index corresponding to the syndrome in the lookup table A, then the error pattern e_m is obtained by index to find in lookup table B. Addition e_m

by one bit at pth position, obtaining error pattern e_m (i.e., em emep), go to step 10.

(9) Look for the index corresponding to the syndrome in the lookup table A, then error pattern e_m is obtained by index to find in the lookup table B.

(10) Subtract the error pattern e_m from receive vector r_m, obtaining the message block e_m, go to stop.

C.2 Decoding Process of the LTD Algorithm in Decoding (41, 21, 9)

1) Initialized by letting p=0.

2) Compute the syndromes S₁ for a received vector r.

3) If S₁=0 go to stop.

4) If syndrome is in T, go to step 9.

5) Check S₁+^p value in T. If syndrome is in T, go to step 8.

6) Compute p p1.

7) If p n4, go to stop. Otherwise return to step 5.

8) If p k1, look for the index corresponding to the syndrome in the lookup table A, then the error pattern e^'_m is obtained by index to find in lookup table B. Addition e^'_m by one bit at pth position, obtaining error pattern e_m (i.e., em em^' ep where ep is set of binary 21-tuples in which only the p-th coordinate contains a nonzero value at p-th position), go to step 10.

9) Look for the index corresponding to the syndrome in the lookup table A, then error pattern

em is obtained by index to find in the lookup table B.

10) Subtract the error pattern e_m from received vector _r_m, obtaining the message block c_m, go to stop.

C.3 Decoding Process of the LTD Algorithm in Decoding (47, 24, 11)

(1) Initialized by letting p=0.

(2) Compute the syndromes S₁ for a received vector r.

(3) If S₁=0 go to stop.

(4) If the syndrome is in T, go to step 9.

(5) Check S₁+^p value in T. If the syndrome is in T, go to step 8.

(6) Compute p p1.

(7) If p n5, go to stop. Otherwise return to step 5.

(8) If p k1, look for the index corresponding to the syndrome in the lookup table A, then the error pattern e is obtained by the index to be found in the lookup table B. Addition _m^'

e by one bit at pth position, obtaining error pattern m e (i.e., _m em em^' ep), go to step 10.

(9) Look for the index corresponding to the syndrome in the lookup table A, then error pattern e is obtained by index to find in the lookup table B. _m

(10) Subtract the error pattern e from the received vector _m r , obtaining the message _m block c , go to stop. _m

)

Fig.I The flowchart of direct method algorithm

)

Fig. II. LTD algorithm Flowchart of the (n, k, d) QR encoder and decoder.

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