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doi: 10.3934/amc.2022030
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Low-density and high-density asymmetric CT-burst correcting integer codes

Nabin Kumar Pokhrel and  Pankaj Kumar Das ,

Department of Mathematical Sciences, Tezpur University, Napaam, Sonitpur, Assam-784028, India

* Corresponding author: Pankaj Kumar Das

Received  September 2021 Revised  March 2022 Early access April 2022

Fund Project: The first author is supported by the University Grants Commission, India (Ref. No: 1112/(CSIR-UGC NET JUNE 2017))

Two classes of integer codes correcting low-density and high-density asymmetric CT-bursts within a $ b $-bit byte have been presented here. Unlike the previously studied integer codes correcting burst errors, here we study such codes with weight constraint on the burst. The proposed codes are compared with similar integer codes in terms of various properties, viz. memory consumption and number of table look ups. This article winds up with an idea of determining the probability of erroneous decoding and an approach for undetected error probability.

Keywords: Integer codes, asymmetric CT-bursts, syndromes, look up table, quad core processor, probability.
Mathematics Subject Classification: Primary: 94B35, 94B60, 94B65.
Citation: Nabin Kumar Pokhrel, Pankaj Kumar Das. Low-density and high-density asymmetric CT-burst correcting integer codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022030
References:
[1]

I. F. Blake, Codes over certain rings, Information and Control, 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.

[2]

I. F. Blake, Codes over integer residue rings, Information and Control, 29 (1975), 295-300.  doi: 10.1016/S0019-9958(75)80001-5.

[3]

B. Buccimazza, B. K. Dass and S. Jain, High-density-burst error detection, Journal of Discrete Mathematical Sciences and Cryptography, 7 (2004), 5-21.  doi: 10.1080/09720529.2004.10697984.

[4]

D. Burton, Elementary Number Theory, 7$^{th}$ Edition, Mcgraw-Hill, 2011.

[5]

R. T. Chien and D. T. Tang, On definitions of a burst, IBM Journal of Research and Development, 9 (1965), 292-293.  doi: 10.1147/rd.94.0292.

[6]

P. K. Das and N. K. Pokhrel, Asymmetric CT-burst correcting integer codes, 2021 5th International Conference on Information Systems and Computer Networks (ISCON), Mathura, India, (2021), 1–5.

[7]

B. K. Dass, F. Eugeni and S. Innamorati, Low-Density burst error locating/correcting linear codes, Journal of Information and Optimization Sciences, 16 (1995), 533-548.  doi: 10.1080/02522667.1995.10699249.

[8]

B. K. Dass, G. Sobha and M. Zannetti, High-density CT burst error-locating linear codes, Journal of Interdisciplinary Mathematics, 5 (2002), 243-249.  doi: 10.1080/09720502.2002.10700320.

[9]

P. Fire, A Class of Multiple-Error-Correcting Binary Codes for Non-Independent Errors, Sylvania Report RSL-E-2, Sylvania Reconnaissance Systems Laboratory, Mountain View, California, 1959.

[10]

T. Klove, Error Correcting Codes for the Asymmetric Channel, Technical Report $18-09-07-81$, Norway: Department of Informatics, University of Bergen, 1981.

[11]

H. Kostadinov and N. Manev, Integer codes correcting asymmetric errors in nand flash memories, Mathematics, 9 (2021), 1269.  doi: 10.3390/math9111269.

[12]

H. Kostadinov, H. Morita and N. Manev, Derivation on bit error probability of coded QAM using integer codes, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E87-A (2004), 3397-3403. 

[13]

H. Kostadinov, H. Morita and N. Manev, On ($\pm$1) error correctable integer codes, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E93-A (2010), 2758-2761. 

[14]

V. I. Levenshteĭn and A. J. H. Vinck, Perfect (d, k) codes capable of correcting single peak-shifts, IEEE Transactions on Information Theory, 39 (1993), 656-662.  doi: 10.1109/18.212300.

[15]

K. Mehlhorn and P. Sanders, Algorithms and Data Structures: The Basic Toolbox, Springer-Verlag, Berlin, 2008.

[16]

S. Park and B. Bose, Burst asymmetric/unidirectional error correcting/detecting codes, Digest of Papers. Fault-Tolerant Computing: 20th International Symposium, (1992), 273–280.

[17] W. W. Peterson and E. J. Weldon. Jr., Error-Correcting Codes, Second edition, The M.I.T. Press, Cambridge, Mass.-London, 1972. 
[18]

A. Radonjic, Integer codes correcting single asymmetric errors, Annals of Telecommunications, 76 (2021), 109-113. 

[19]

A. Radonjic and V. Vujicic, Integer codes correcting burst errors within a byte, IEEE Transactions on Computers, 62 (2013), 411-415.  doi: 10.1109/TC.2011.243.

[20]

A. Radonjic and V. Vujicic, Integer codes correcting spotty byte asymmetric errors, IEEE Communications Letters, 20 (2016), 2338-2341. 

[21]

A. Radonjic and V. Vujicic, Integer codes correcting single errors and burst asymmetric errors within a byte, Inform. Process. Lett., 121 (2017), 45-50.  doi: 10.1016/j.ipl.2017.01.010.

[22]

A. Radonjic and V. Vujicic, Integer codes correcting burst and random asymmetric errors within a byte, Journal of the Franklin Institute, 355 (2018), 981-996.  doi: 10.1016/j.jfranklin.2017.11.033.

[23]

A. Radonjic and V. Vujicic, Integer codes correcting burst asymmetric errors within a byte and double asymmetric errors, Cryptography and Communications, 12 (2020), 221-230.  doi: 10.1007/s12095-019-00388-0.

[24]

R. R. Varshamov and G. M. Tenengolz, One asymmetrical error-correcting codes, Avtomatika i Telemehanika, 26 (1965), 288-292. 

[25]

A. J. H. Vinck and H. Morita, Codes over the ring of integer modulom, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 81A (1998), 2013-2018. 

[26]

A. D. Wyner, Low-density-burst-correcting codes, IEEE Transactions on Information Theory, IT-9 (1963), 124. 

show all references

References:
[1]

I. F. Blake, Codes over certain rings, Information and Control, 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.

[2]

I. F. Blake, Codes over integer residue rings, Information and Control, 29 (1975), 295-300.  doi: 10.1016/S0019-9958(75)80001-5.

[3]

B. Buccimazza, B. K. Dass and S. Jain, High-density-burst error detection, Journal of Discrete Mathematical Sciences and Cryptography, 7 (2004), 5-21.  doi: 10.1080/09720529.2004.10697984.

[4]

D. Burton, Elementary Number Theory, 7$^{th}$ Edition, Mcgraw-Hill, 2011.

[5]

R. T. Chien and D. T. Tang, On definitions of a burst, IBM Journal of Research and Development, 9 (1965), 292-293.  doi: 10.1147/rd.94.0292.

[6]

P. K. Das and N. K. Pokhrel, Asymmetric CT-burst correcting integer codes, 2021 5th International Conference on Information Systems and Computer Networks (ISCON), Mathura, India, (2021), 1–5.

[7]

B. K. Dass, F. Eugeni and S. Innamorati, Low-Density burst error locating/correcting linear codes, Journal of Information and Optimization Sciences, 16 (1995), 533-548.  doi: 10.1080/02522667.1995.10699249.

[8]

B. K. Dass, G. Sobha and M. Zannetti, High-density CT burst error-locating linear codes, Journal of Interdisciplinary Mathematics, 5 (2002), 243-249.  doi: 10.1080/09720502.2002.10700320.

[9]

P. Fire, A Class of Multiple-Error-Correcting Binary Codes for Non-Independent Errors, Sylvania Report RSL-E-2, Sylvania Reconnaissance Systems Laboratory, Mountain View, California, 1959.

[10]

T. Klove, Error Correcting Codes for the Asymmetric Channel, Technical Report $18-09-07-81$, Norway: Department of Informatics, University of Bergen, 1981.

[11]

H. Kostadinov and N. Manev, Integer codes correcting asymmetric errors in nand flash memories, Mathematics, 9 (2021), 1269.  doi: 10.3390/math9111269.

[12]

H. Kostadinov, H. Morita and N. Manev, Derivation on bit error probability of coded QAM using integer codes, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E87-A (2004), 3397-3403. 

[13]

H. Kostadinov, H. Morita and N. Manev, On ($\pm$1) error correctable integer codes, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, E93-A (2010), 2758-2761. 

[14]

V. I. Levenshteĭn and A. J. H. Vinck, Perfect (d, k) codes capable of correcting single peak-shifts, IEEE Transactions on Information Theory, 39 (1993), 656-662.  doi: 10.1109/18.212300.

[15]

K. Mehlhorn and P. Sanders, Algorithms and Data Structures: The Basic Toolbox, Springer-Verlag, Berlin, 2008.

[16]

S. Park and B. Bose, Burst asymmetric/unidirectional error correcting/detecting codes, Digest of Papers. Fault-Tolerant Computing: 20th International Symposium, (1992), 273–280.

[17] W. W. Peterson and E. J. Weldon. Jr., Error-Correcting Codes, Second edition, The M.I.T. Press, Cambridge, Mass.-London, 1972. 
[18]

A. Radonjic, Integer codes correcting single asymmetric errors, Annals of Telecommunications, 76 (2021), 109-113. 

[19]

A. Radonjic and V. Vujicic, Integer codes correcting burst errors within a byte, IEEE Transactions on Computers, 62 (2013), 411-415.  doi: 10.1109/TC.2011.243.

[20]

A. Radonjic and V. Vujicic, Integer codes correcting spotty byte asymmetric errors, IEEE Communications Letters, 20 (2016), 2338-2341. 

[21]

A. Radonjic and V. Vujicic, Integer codes correcting single errors and burst asymmetric errors within a byte, Inform. Process. Lett., 121 (2017), 45-50.  doi: 10.1016/j.ipl.2017.01.010.

[22]

A. Radonjic and V. Vujicic, Integer codes correcting burst and random asymmetric errors within a byte, Journal of the Franklin Institute, 355 (2018), 981-996.  doi: 10.1016/j.jfranklin.2017.11.033.

[23]

A. Radonjic and V. Vujicic, Integer codes correcting burst asymmetric errors within a byte and double asymmetric errors, Cryptography and Communications, 12 (2020), 221-230.  doi: 10.1007/s12095-019-00388-0.

[24]

R. R. Varshamov and G. M. Tenengolz, One asymmetrical error-correcting codes, Avtomatika i Telemehanika, 26 (1965), 288-292. 

[25]

A. J. H. Vinck and H. Morita, Codes over the ring of integer modulom, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 81A (1998), 2013-2018. 

[26]

A. D. Wyner, Low-density-burst-correcting codes, IEEE Transactions on Information Theory, IT-9 (1963), 124. 

Figure 1.  Z-Channel
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Figure 2.  Width presentation of an encoded codeword
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Figure 3.  Width of each syndrome entry
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Figure 4.  Code rate vs BER and probability (low-density)
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Figure 6.  Code rate vs BER and probability (high-density)
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Table 1.  $LUT_2$ for integer $(16, 8) LACTB_{(3/6, 8)}C$ code
Sl.No. Syndrome ($S_1$) Error Loc.($i$) Error($e$) Sl.No. Syndrome($S_1$) Error Loc.($i$) Error($e$)
1 1 2 1 49 107 1 74
2 2 2 2 50 115 1 70
3 3 2 3 51 118 1 196
4 4 2 4 52 119 1 68
5 5 2 5 53 123 1 66
6 6 2 6 54 132 2 132
7 7 2 7 55 140 2 140
8 9 2 9 56 148 2 148
9 10 2 10 57 151 1 52
10 11 2 11 58 155 1 50
11 12 2 12 59 157 1 49
12 13 2 13 60 164 2 164
13 14 2 14 61 167 1 44
14 17 2 17 62 171 1 42
15 18 2 18 63 173 1 41
16 19 2 19 64 179 1 38
17 20 2 20 65 181 1 37
18 21 2 21 66 182 1 164
19 22 2 22 67 183 1 36
20 25 2 25 68 185 1 35
21 26 2 26 69 187 1 34
22 28 2 28 70 189 1 33
23 33 2 33 71 196 2 196
24 34 2 34 72 199 1 28
25 35 2 35 73 203 1 26
26 36 2 36 74 205 1 25
27 37 2 37 75 211 1 22
28 38 2 38 76 213 1 21
29 41 2 41 77 214 1 148
30 42 2 42 78 215 1 20
31 44 2 44 79 217 1 19
32 49 2 49 80 219 1 18
33 50 2 50 81 221 1 17
34 52 2 52 82 227 1 14
35 55 1 100 83 229 1 13
36 59 1 98 84 230 1 140
37 66 2 66 85 231 1 12
38 68 2 68 86 233 1 11
39 70 2 70 87 235 1 10
40 74 2 74 88 237 1 9
41 76 2 76 89 241 1 7
42 82 2 82 90 243 1 6
43 84 2 84 91 245 1 5
44 87 1 84 92 246 1 132
45 91 1 82 93 247 1 4
46 98 2 98 94 249 1 3
47 100 2 100 95 251 1 2
48 103 1 76 96 253 1 1
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Sl.No. Syndrome ($S_1$) Error Loc.($i$) Error($e$) Sl.No. Syndrome($S_1$) Error Loc.($i$) Error($e$)
1 1 2 1 49 107 1 74
2 2 2 2 50 115 1 70
3 3 2 3 51 118 1 196
4 4 2 4 52 119 1 68
5 5 2 5 53 123 1 66
6 6 2 6 54 132 2 132
7 7 2 7 55 140 2 140
8 9 2 9 56 148 2 148
9 10 2 10 57 151 1 52
10 11 2 11 58 155 1 50
11 12 2 12 59 157 1 49
12 13 2 13 60 164 2 164
13 14 2 14 61 167 1 44
14 17 2 17 62 171 1 42
15 18 2 18 63 173 1 41
16 19 2 19 64 179 1 38
17 20 2 20 65 181 1 37
18 21 2 21 66 182 1 164
19 22 2 22 67 183 1 36
20 25 2 25 68 185 1 35
21 26 2 26 69 187 1 34
22 28 2 28 70 189 1 33
23 33 2 33 71 196 2 196
24 34 2 34 72 199 1 28
25 35 2 35 73 203 1 26
26 36 2 36 74 205 1 25
27 37 2 37 75 211 1 22
28 38 2 38 76 213 1 21
29 41 2 41 77 214 1 148
30 42 2 42 78 215 1 20
31 44 2 44 79 217 1 19
32 49 2 49 80 219 1 18
33 50 2 50 81 221 1 17
34 52 2 52 82 227 1 14
35 55 1 100 83 229 1 13
36 59 1 98 84 230 1 140
37 66 2 66 85 231 1 12
38 68 2 68 86 233 1 11
39 70 2 70 87 235 1 10
40 74 2 74 88 237 1 9
41 76 2 76 89 241 1 7
42 82 2 82 90 243 1 6
43 84 2 84 91 245 1 5
44 87 1 84 92 246 1 132
45 91 1 82 93 247 1 4
46 98 2 98 94 249 1 3
47 100 2 100 95 251 1 2
48 103 1 76 96 253 1 1
Table 2.  $LUT_2$ for an integer $HACTB_{(2/4, 8)}C$ code
Sl.No. Syndrome($S_2$) Error Loc.($i$) Error($e$) Sl.No. Syndrome($S_2$) Error Loc.($i$) Error($e$)
1 3 2 3 36 76 1 14
2 5 2 5 37 77 1 88
3 6 2 6 38 80 2 80
4 7 2 7 39 83 1 22
5 9 2 9 40 88 2 88
6 10 2 10 41 90 1 30
7 11 2 11 42 91 1 104
8 12 2 12 43 98 1 112
9 13 2 13 44 100 1 5
10 14 2 14 45 104 2 104
11 15 2 15 46 105 1 120
12 18 2 18 47 107 1 13
13 20 2 20 48 112 2 112
14 21 1 24 49 120 2 120
15 22 2 22 50 126 1 144
16 24 2 24 51 138 1 12
17 26 2 26 52 144 2 144
18 28 2 28 53 145 1 20
19 30 2 30 54 152 1 28
20 35 1 40 55 154 1 176
21 36 2 36 56 159 1 36
22 38 1 7 57 162 1 3
23 40 2 40 58 166 1 44
24 42 1 48 59 169 1 11
25 44 2 44 60 173 1 52
26 45 1 15 61 176 2 176
27 48 2 48 62 180 1 60
28 49 1 56 63 182 1 208
29 52 2 52 64 200 1 10
30 56 2 56 65 207 1 18
31 60 2 60 66 208 2 208
32 63 1 72 67 210 1 240
33 69 1 6 68 214 1 26
34 70 1 80 69 231 1 9
35 72 2 72 70 240 2 240
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Sl.No. Syndrome($S_2$) Error Loc.($i$) Error($e$) Sl.No. Syndrome($S_2$) Error Loc.($i$) Error($e$)
1 3 2 3 36 76 1 14
2 5 2 5 37 77 1 88
3 6 2 6 38 80 2 80
4 7 2 7 39 83 1 22
5 9 2 9 40 88 2 88
6 10 2 10 41 90 1 30
7 11 2 11 42 91 1 104
8 12 2 12 43 98 1 112
9 13 2 13 44 100 1 5
10 14 2 14 45 104 2 104
11 15 2 15 46 105 1 120
12 18 2 18 47 107 1 13
13 20 2 20 48 112 2 112
14 21 1 24 49 120 2 120
15 22 2 22 50 126 1 144
16 24 2 24 51 138 1 12
17 26 2 26 52 144 2 144
18 28 2 28 53 145 1 20
19 30 2 30 54 152 1 28
20 35 1 40 55 154 1 176
21 36 2 36 56 159 1 36
22 38 1 7 57 162 1 3
23 40 2 40 58 166 1 44
24 42 1 48 59 169 1 11
25 44 2 44 60 173 1 52
26 45 1 15 61 176 2 176
27 48 2 48 62 180 1 60
28 49 1 56 63 182 1 208
29 52 2 52 64 200 1 10
30 56 2 56 65 207 1 18
31 60 2 60 66 208 2 208
32 63 1 72 67 210 1 240
33 69 1 6 68 214 1 26
34 70 1 80 69 231 1 9
35 72 2 72 70 240 2 240
Table 3.  First 32 possible coefficients for integer $LACTB_{({d/l, b})}C$ codes
$b$ ${l}$ $\lfloor \frac{{l}}{2} \rfloor$ Coefficients
7 5 2 2, 33, 47,100
8 4 2 2
8 6 3 2
10 4 2 2, 7, 13, 15, 23, 37, 41, 47, 49, 83
10 6 3 2,135
10 7 3 2
16 4 2 2, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 81, 83, 89, 91, 97,101,105,107,109
16 5 2 2, 7, 11, 13, 15, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 67, 71, 73, 77, 79, 81, 83, 89, 97,101,105,107,109,121,125,127
16 6 3 2, 15, 23, 29, 31, 43, 47, 53, 59, 67, 71, 73, 77, 79, 83, 89, 97,101,107,117,131,137,139,149,157,163,167,181,199,227,233,251
16 8 4 2, 31, 61,207,776, 7769
16 9 4 2, 31,413, 1536, 16904
32 6 3 2, 15, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 81, 83, 89, 97,101,103,107,109,113,127,131,137,139,149,151,157,163
32 7 3 2, 15, 29, 31, 43, 47, 53, 59, 61, 71, 77, 79, 83, 89,101,103,107,109,113,117,127,131,137,139,149,151,157,163,167,173,179,181
32 8 4 2, 31, 61, 63, 79, 95,103,107,121,127,151,157,167,173,179,181,191,199,211,221,223,227,229,233,239,241,251,257,263,269,271,277
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$b$ ${l}$ $\lfloor \frac{{l}}{2} \rfloor$ Coefficients
7 5 2 2, 33, 47,100
8 4 2 2
8 6 3 2
10 4 2 2, 7, 13, 15, 23, 37, 41, 47, 49, 83
10 6 3 2,135
10 7 3 2
16 4 2 2, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 81, 83, 89, 91, 97,101,105,107,109
16 5 2 2, 7, 11, 13, 15, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 67, 71, 73, 77, 79, 81, 83, 89, 97,101,105,107,109,121,125,127
16 6 3 2, 15, 23, 29, 31, 43, 47, 53, 59, 67, 71, 73, 77, 79, 83, 89, 97,101,107,117,131,137,139,149,157,163,167,181,199,227,233,251
16 8 4 2, 31, 61,207,776, 7769
16 9 4 2, 31,413, 1536, 16904
32 6 3 2, 15, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 81, 83, 89, 97,101,103,107,109,113,127,131,137,139,149,151,157,163
32 7 3 2, 15, 29, 31, 43, 47, 53, 59, 61, 71, 77, 79, 83, 89,101,103,107,109,113,117,127,131,137,139,149,151,157,163,167,173,179,181
32 8 4 2, 31, 61, 63, 79, 95,103,107,121,127,151,157,167,173,179,181,191,199,211,221,223,227,229,233,239,241,251,257,263,269,271,277
Table 4.  First 32 possible coefficients for integer $HACTB_{({h/l, b})}C$ codes
$b$ ${l}$ $\lceil \frac{{l}}{2} \rceil$ Coefficients
8 3 2 2, 3, 29, 37
8 4 2 31
8 5 3 239
10 4 2 2, 13, 41
10 6 3 991
10 5 3 7
16 5 3 2, 5, 11, 17, 35, 37, 39, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 83, 97,101,107,113,119,127,131,137,149,151,157,163,169,173
16 6 3 2, 11, 67, 71, 73, 79, 95,103,129,137,179,193,217,267,293,311,327,373,389,393,449,461,517,725,761, 1001, 2501, 2527, 2999, 3481, 3643, 4517
16 7 4 2, 9, 43,131,139,163,183,197,199,209,251,491, 2477, 4727
16 8 4 7, 61, 22447
16 9 5 2389, 21769, 65279
32 6 3 2, 11, 17, 65, 67, 69, 71, 73, 79, 83, 89, 97,101,103,107,109,113,121,127,131,133,137,139,149,151,157,163,167,173,179,181,187
32 8 4 2, 19, 87, 97,131,137,161,193,257,263,265,269,271,277,281,283,289,293,307,311,313,317,331,337,341,347,349,353,359,361,367,373
32 9 5 2, 17, 47, 77,129,131,139,193,197,257,263,265,269,277,281,289,293,321,337,353,389,401,449,521,523,529,531,533,541,547,551,557
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$b$ ${l}$ $\lceil \frac{{l}}{2} \rceil$ Coefficients
8 3 2 2, 3, 29, 37
8 4 2 31
8 5 3 239
10 4 2 2, 13, 41
10 6 3 991
10 5 3 7
16 5 3 2, 5, 11, 17, 35, 37, 39, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 83, 97,101,107,113,119,127,131,137,149,151,157,163,169,173
16 6 3 2, 11, 67, 71, 73, 79, 95,103,129,137,179,193,217,267,293,311,327,373,389,393,449,461,517,725,761, 1001, 2501, 2527, 2999, 3481, 3643, 4517
16 7 4 2, 9, 43,131,139,163,183,197,199,209,251,491, 2477, 4727
16 8 4 7, 61, 22447
16 9 5 2389, 21769, 65279
32 6 3 2, 11, 17, 65, 67, 69, 71, 73, 79, 83, 89, 97,101,103,107,109,113,121,127,131,133,137,139,149,151,157,163,167,173,179,181,187
32 8 4 2, 19, 87, 97,131,137,161,193,257,263,265,269,271,277,281,283,289,293,307,311,313,317,331,337,341,347,349,353,359,361,367,373
32 9 5 2, 17, 47, 77,129,131,139,193,197,257,263,265,269,277,281,289,293,321,337,353,389,401,449,521,523,529,531,533,541,547,551,557
Table 5.  Lookup sizes for $LACTB_{({d/l, b})}C$ codes
Codes $b$ ${l}$ $ \lfloor \frac{{l}}{2} \rfloor$ $LUT_1$ size $LUT_2$ size No of table look ups
(144,128) 16 4 2 4$\times$ 16B 2.11 KB 1 $\leq \eta_{TL} \leq$ 10
(528,512) 16 5 2 4$\times$ 64B 9.41 KB 1 $\leq \eta_{TL} \leq$ 12
(512,480) 32 6 3 4$\times$ 60B 58.75 KB 1 $\leq \eta_{TL} \leq$ 14
(1056, 1024) 32 8 4 4$\times$ 128B 0.46 MB 1 $\leq \eta_{TL} \leq$ 17
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Codes $b$ ${l}$ $ \lfloor \frac{{l}}{2} \rfloor$ $LUT_1$ size $LUT_2$ size No of table look ups
(144,128) 16 4 2 4$\times$ 16B 2.11 KB 1 $\leq \eta_{TL} \leq$ 10
(528,512) 16 5 2 4$\times$ 64B 9.41 KB 1 $\leq \eta_{TL} \leq$ 12
(512,480) 32 6 3 4$\times$ 60B 58.75 KB 1 $\leq \eta_{TL} \leq$ 14
(1056, 1024) 32 8 4 4$\times$ 128B 0.46 MB 1 $\leq \eta_{TL} \leq$ 17
Table 6.  Lookup sizes for $HACTB_{({h/l, b})}C$ codes
Codes $b$ ${l}$ $\lceil \frac{{l}}{2}\rceil $ $LUT_1$ size $LUT_2$ size No of table look ups
$(240,224)$ $16$ $7$ $4$ $4 \times 28$ B $28.35$ KB $1 \leq \eta_{TL} \leq 14$
$(512,496)$ $16$ $6$ $3$ $4 \times 62$ B $42.33$ KB $1 \leq \eta_{TL} \leq 15$
$(544,512)$ $32$ $6$ $3$ $4 \times 64$ B $0.1$ MB $1 \leq \eta_{TL} \leq 15$
$(1056, 1024)$ $32$ $8$ $4$ $4 \times 128$ B $0.71$ MB $1 \leq \eta_{TL} \leq 18$
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Codes $b$ ${l}$ $\lceil \frac{{l}}{2}\rceil $ $LUT_1$ size $LUT_2$ size No of table look ups
$(240,224)$ $16$ $7$ $4$ $4 \times 28$ B $28.35$ KB $1 \leq \eta_{TL} \leq 14$
$(512,496)$ $16$ $6$ $3$ $4 \times 62$ B $42.33$ KB $1 \leq \eta_{TL} \leq 15$
$(544,512)$ $32$ $6$ $3$ $4 \times 64$ B $0.1$ MB $1 \leq \eta_{TL} \leq 15$
$(1056, 1024)$ $32$ $8$ $4$ $4 \times 128$ B $0.71$ MB $1 \leq \eta_{TL} \leq 18$
Table 7.  Comparison of some integer codes with $32$ information bytes
Codes Error pattern $b$ ${l}$ $LUT_2$ size $\#$ table look ups
$LACTB_{(\textit{d/l, }b)}C$ Proposed low-density 32 8 0.46 MB 1 $\leq \eta_{TL} \leq $ 17
$HACTB_{(\textit{h/l}, b)}C$ Proposed high-density 32 8 0.71 MB 1 $\leq \eta_{TL} \leq $ 18
$(CT_{\textit{l}}B)_b$ from [6] Asymmetric CT-bursts 32 8 0.96 MB 1 $\leq \eta_{TL} \leq $ 18
From [19] Symmetric bursts 32 8 3.84 MB 1 $\leq \eta_{TL} \leq $ 20
From [22] Bursts and random asymmetric 32 8 2.32 MB 1 $\leq \eta_{TL} \leq $ 20
From [23] Asymmetric bursts and double random asymmetric 32 8 8.91 MB 1 $\leq \eta_{TL} \leq $ 21
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Codes Error pattern $b$ ${l}$ $LUT_2$ size $\#$ table look ups
$LACTB_{(\textit{d/l, }b)}C$ Proposed low-density 32 8 0.46 MB 1 $\leq \eta_{TL} \leq $ 17
$HACTB_{(\textit{h/l}, b)}C$ Proposed high-density 32 8 0.71 MB 1 $\leq \eta_{TL} \leq $ 18
$(CT_{\textit{l}}B)_b$ from [6] Asymmetric CT-bursts 32 8 0.96 MB 1 $\leq \eta_{TL} \leq $ 18
From [19] Symmetric bursts 32 8 3.84 MB 1 $\leq \eta_{TL} \leq $ 20
From [22] Bursts and random asymmetric 32 8 2.32 MB 1 $\leq \eta_{TL} \leq $ 20
From [23] Asymmetric bursts and double random asymmetric 32 8 8.91 MB 1 $\leq \eta_{TL} \leq $ 21
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