American Institute of Mathematical Sciences

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

 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.

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.

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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.
Z-Channel
Width presentation of an encoded codeword
Width of each syndrome entry
Code rate vs BER and probability (low-density)
Code rate vs BER and probability (high-density)
$LUT_2$ 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
 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
$LUT_2$ 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
 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
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
 $b$ ${l}$ $\lfloor \frac{{l}}{2} \rfloor$ 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
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
 $b$ ${l}$ $\lceil \frac{{l}}{2} \rceil$ 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
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
 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
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 $1 \leq \eta_{TL} \leq 18$
 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 $1 \leq \eta_{TL} \leq 18$
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
 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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