# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022025
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## Constructions of optimal multiply constant-weight codes MCWC$(3,n_1;1,n_2;1,n_3;8)s$

 1 School of Sciences, Nantong University, Nantong 226007, China 2 School of Mathematics and Statistics, University of New South Wales, N.S.W. 2052, Australia

* Corresponding author: Jinhua Wang

Received  November 2021 Revised  March 2022 Early access April 2022

Fund Project: The third author's research is supported by the National Natural Science Foundation of China under Grant No.11671402

A binary code ${\mathcal{C}}$ of length $n = \sum_{i = 1}^{m}n_i$ and minimum distance $d$ is said to be of multiply constant-weight and denoted by MCWC$(w_1, n_1$; $w_2, n_2$; $\ldots$; $w_m, n_m$; $d)$, if each codeword has weight $w_1$ in the first $n_1$ coordinates, weight $w_2$ in the next $n_2$ coordinates, and so on and so forth. Multiply constant-weight codes (MCWCs) can be utilized to improve the reliability of certain physically unclonable function responses and have been widely studied. Research showed that multiply constant-weight codes are equivalent to generalized packing designs and generalized Howell designs (GHDs) can be regarded as generalized packing designs with a special block type. In this paper, we give combinatorial constructions for optimal MCWC$(3, n_1;1, n_2;1, n_3;8)$s by a class of generalized packing designs, which come from generalized Howell designs. Furthermore, for $e = 3, 4, 5$, we prove that there exists a GHD $(n+e, 3n)$ if and only if $n\ge 2e+1$ leaving some possible exceptions.

Citation: Liying Pan, R. Julian R. Abel, Jinhua Wang. Constructions of optimal multiply constant-weight codes MCWC$(3,n_1;1,n_2;1,n_3;8)s$. Advances in Mathematics of Communications, doi: 10.3934/amc.2022025
##### References:
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Arhin, On the Construction and Structure of SOMAs and Related Partial Linear Spaces, Ph.D. Thesis, University of London, 2006. [7] J. Arhin, Every SOMA$(n-2, n)$ is Trojan, Discrete Math., 310 (2010), 303-311.  doi: 10.1016/j.disc.2008.09.050. [8] R. F. Bailey and A. C. Burgess, Generalized packing designs, Discrete Math., 313 (2013), 1167-1190.  doi: 10.1016/j.disc.2011.11.039. [9] E. F. Brickell, A few results in message authentication, Congr.Numer., 43 (1984), 141-154. [10] P. J. Cameron, A generalisation of $t$-designs, Discrete Math., 309 (2009), 4835-4842.  doi: 10.1016/j.disc.2008.07.005. [11] Y. M. Chee, Z. Cherif, J. L. Danger, S. Guilley, H. M. Kiah, J. L. Kim, P. Solé and X. Zhang, Multiply constant-weight codes and the reliability of loop physically unclonable functions, IEEE Trans. Inf. Theory, 60 (2014), 7026-7034.  doi: 10.1109/TIT.2014.2359207. [12] Y. M. Chee, H. M. Kiah, H. Zhang and X. Zhang, Constructions of optimal and near optimal multiply constant-weight codes, IEEE Trans. Inf. Theory, 63 (2017), 3621-3629.  doi: 10.1109/TIT.2017.2690450. [13] Z. Cherif, J. L. Danger, S. Guilley, J. L. Kim and P. Solé, Multiply constant weight codes, In IEEE International Symposium on Information Theory, (2013). doi: 10.1109/ISIT.2013.6620237. [14] Z. Cherif, J. L. Danger, S. Guilley and L. Bossuet, An easy-to-design puf based on a single oscillator: The loop puf, In Proceedings of the 2012 15th Euromicro Conference on Digital System Design, ser. DSD'12, (2012), 156–162. doi: 10.1109/DSD.2012.22. [15] C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, 1996.  doi: 10.1201/9781420049954. [16] C. J. Colbourn, E. R. Lamken, A. C. H. Ling and W. H. Mills, The existence of Kirkman squares-doubly resolvable $(v, 3, 1)$-BIBDs, Des. Codes Cryptogr., 26 (2002), 169-196.  doi: 10.1023/A:1016513527747. [17] M. Deza and S. A. Vanstone, Bounds for permutation arrays, J. Statist. Plan. Inference, 2 (1978), 197-209.  doi: 10.1016/0378-3758(78)90008-3. [18] J. H. Dinitz and D. R. Stinson, Room squares and related designs, In Contemporary Design Theory, (eds. J.H. Dinitz and D. R. Stinson (eds.), Wiley, New York, (1992), 137–204. [19] J. Du, R. J. R. Abel and J. Wang, Some new resolvable GDDs with $k = 4$ and doubly resolvable GDDs with $k = 3$, Discrete Math., 338 (2015), 2015-2118.  doi: 10.1016/j.disc.2015.05.008. [20] T. Etzion, Optimal doubly constant weight codes, J. Combin. Des., 16 (2008), 137-151.  doi: 10.1002/jcd.20160. [21] B. Gassend, D. Clarke, M. van Dijk and S. Devadas, Silicon physical random functions, In Proceedings of the 9th ACM Conference on Computer and Communications Security, ser. CCS'02, (2002), 148–160. doi: 10.1145/586110.586132. [22] R. Pappu, B. Recht, J. Taylor and N. Gershenfeld, Physical one-way functions, Science, 297 (2002), 2026-2030.  doi: 10.1126/science.1074376. [23] E. R. Lamken, The existence of doubly resolvable $(v, 3, 2)$-BIBDs, J. Combin.Theory Ser. A, 72 (1995), 50-76.  doi: 10.1016/0097-3165(95)90028-4. [24] R. C. Mullin and W. D. Wallis, The existence of Room squares, Aequ. Math., 13 (1975), 1-7.  doi: 10.1007/BF01834113. [25] N. C. K. Phillips and W. D. Wallis, All solutions to a tournament problem, Congr. Numer., 114 (1996), 193-196. [26] J. Schönheim, On maximal systems of $k$-tuples, Studia Sci. Math. Hungar., 1 (1966), 363-368. [27] L. H. Soicher, On the structure and classification of SOMAs: Generalizations of mutually orthogonal Latin squares, Electron. J. Combin., 6 (1999), 32, 15 pp. [28] D. R. Stinson, Some Classes of Frames, and the Spectra of Skew Room Squares and Howell Designs, Ph.D. Thesis, University of Waterloo, 1981. [29] D. R. Stinson, The existence of Howell designs of odd side, J. Combin. Theory Ser. A, 32 (1982), 53-65.  doi: 10.1016/0097-3165(82)90064-4. [30] G. E. Suh and S. Devadas, Physical unclonable functions for device authentication and secret key generation, In Proceedings of the 44th Annual Design Automation Conference, ser. DAC'07, (2007), 9–14. [31] D. T. Todorov, Four mutually orthogonal Latin squares of order $14$, J. Combin. Des., 20 (2012), 363-367.  doi: 10.1002/jcd.21298. [32] C. Wang, Y. Chang and T. Feng, Optimal multiply constant-weight codes from generalized Howell designs, Graphs Combin., 35 (2019), 611-632.  doi: 10.1007/s00373-019-02020-7. [33] C. Wang and B. Du, Existence of generalized Howell designs of side $n+ 1$, Util. Math., 80 (2009), 143-159. [34] X. Wang, H. Wei, C. Shangguan and G. Ge, New bounds and constructions for multiply constant-weight codes, IEEE Trans. Inf. Theory, 62 (2016), 6315-6327.  doi: 10.1109/TIT.2016.2609389. [35] J. Yao, Y. Hu and J. Wang, Existence of generalized Howell designs GHD$(n+5, 3n)$, J. Guangxi Norm. Uni. (Natural Science Edition), 39 (2021), 103-114.

show all references

##### References:
 [1] R. J. R. Abel, Existence of five MOLS of orders 18 and 60, J. Combin. Des., 23 (2015), 135-139.  doi: 10.1002/jcd.21384. [2] R. J. R. Abel, R. F. Bailey, A. C. Burgess, P. Danziger and E. Mendelsohn, On generalized Howell designs with block size three, Des. Codes Cryptogr., 81 (2016), 365-391.  doi: 10.1007/s10623-015-0162-7. [3] R. J. R. Abel, N. Chan, C. J. Colbourn, E. R. Lamken, C. Wang and J. Wang, Doubly resolvable nearly Kirkman triple systems, J. Combin. Des., 21 (2013), 342-358.  doi: 10.1002/jcd.21342. [4] R. J. R. Abel, E. R. Lamken and J. Wang, A few more Kirkman squares and doubly near resolvable BIBDs with block size $3$, Discrete Math., 308 (2008), 1102-1123.  doi: 10.1016/j.disc.2007.04.001. [5] B. A. Anderson, P. J. Schellenberg and D. R. Stinson, The existence of Howell designs of even side, J. Comb. Theory Ser. A, 36 (1984), 23-55.  doi: 10.1016/0097-3165(84)90076-1. [6] J. Arhin, On the Construction and Structure of SOMAs and Related Partial Linear Spaces, Ph.D. Thesis, University of London, 2006. [7] J. Arhin, Every SOMA$(n-2, n)$ is Trojan, Discrete Math., 310 (2010), 303-311.  doi: 10.1016/j.disc.2008.09.050. [8] R. F. Bailey and A. C. Burgess, Generalized packing designs, Discrete Math., 313 (2013), 1167-1190.  doi: 10.1016/j.disc.2011.11.039. [9] E. F. Brickell, A few results in message authentication, Congr.Numer., 43 (1984), 141-154. [10] P. J. Cameron, A generalisation of $t$-designs, Discrete Math., 309 (2009), 4835-4842.  doi: 10.1016/j.disc.2008.07.005. [11] Y. M. Chee, Z. Cherif, J. L. Danger, S. Guilley, H. M. Kiah, J. L. Kim, P. Solé and X. Zhang, Multiply constant-weight codes and the reliability of loop physically unclonable functions, IEEE Trans. Inf. Theory, 60 (2014), 7026-7034.  doi: 10.1109/TIT.2014.2359207. [12] Y. M. Chee, H. M. Kiah, H. Zhang and X. Zhang, Constructions of optimal and near optimal multiply constant-weight codes, IEEE Trans. Inf. Theory, 63 (2017), 3621-3629.  doi: 10.1109/TIT.2017.2690450. [13] Z. Cherif, J. L. Danger, S. Guilley, J. L. Kim and P. Solé, Multiply constant weight codes, In IEEE International Symposium on Information Theory, (2013). doi: 10.1109/ISIT.2013.6620237. [14] Z. Cherif, J. L. Danger, S. Guilley and L. Bossuet, An easy-to-design puf based on a single oscillator: The loop puf, In Proceedings of the 2012 15th Euromicro Conference on Digital System Design, ser. DSD'12, (2012), 156–162. doi: 10.1109/DSD.2012.22. [15] C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, 1996.  doi: 10.1201/9781420049954. [16] C. J. Colbourn, E. R. Lamken, A. C. H. Ling and W. H. Mills, The existence of Kirkman squares-doubly resolvable $(v, 3, 1)$-BIBDs, Des. Codes Cryptogr., 26 (2002), 169-196.  doi: 10.1023/A:1016513527747. [17] M. Deza and S. A. Vanstone, Bounds for permutation arrays, J. Statist. Plan. Inference, 2 (1978), 197-209.  doi: 10.1016/0378-3758(78)90008-3. [18] J. H. Dinitz and D. R. Stinson, Room squares and related designs, In Contemporary Design Theory, (eds. J.H. Dinitz and D. R. Stinson (eds.), Wiley, New York, (1992), 137–204. [19] J. Du, R. J. R. Abel and J. Wang, Some new resolvable GDDs with $k = 4$ and doubly resolvable GDDs with $k = 3$, Discrete Math., 338 (2015), 2015-2118.  doi: 10.1016/j.disc.2015.05.008. [20] T. Etzion, Optimal doubly constant weight codes, J. Combin. Des., 16 (2008), 137-151.  doi: 10.1002/jcd.20160. [21] B. Gassend, D. Clarke, M. van Dijk and S. Devadas, Silicon physical random functions, In Proceedings of the 9th ACM Conference on Computer and Communications Security, ser. CCS'02, (2002), 148–160. doi: 10.1145/586110.586132. [22] R. Pappu, B. Recht, J. Taylor and N. Gershenfeld, Physical one-way functions, Science, 297 (2002), 2026-2030.  doi: 10.1126/science.1074376. [23] E. R. Lamken, The existence of doubly resolvable $(v, 3, 2)$-BIBDs, J. Combin.Theory Ser. A, 72 (1995), 50-76.  doi: 10.1016/0097-3165(95)90028-4. [24] R. C. Mullin and W. D. Wallis, The existence of Room squares, Aequ. Math., 13 (1975), 1-7.  doi: 10.1007/BF01834113. [25] N. C. K. Phillips and W. D. Wallis, All solutions to a tournament problem, Congr. Numer., 114 (1996), 193-196. [26] J. Schönheim, On maximal systems of $k$-tuples, Studia Sci. Math. Hungar., 1 (1966), 363-368. [27] L. H. Soicher, On the structure and classification of SOMAs: Generalizations of mutually orthogonal Latin squares, Electron. J. Combin., 6 (1999), 32, 15 pp. [28] D. R. Stinson, Some Classes of Frames, and the Spectra of Skew Room Squares and Howell Designs, Ph.D. Thesis, University of Waterloo, 1981. [29] D. R. Stinson, The existence of Howell designs of odd side, J. Combin. Theory Ser. A, 32 (1982), 53-65.  doi: 10.1016/0097-3165(82)90064-4. [30] G. E. Suh and S. Devadas, Physical unclonable functions for device authentication and secret key generation, In Proceedings of the 44th Annual Design Automation Conference, ser. DAC'07, (2007), 9–14. [31] D. T. Todorov, Four mutually orthogonal Latin squares of order $14$, J. Combin. Des., 20 (2012), 363-367.  doi: 10.1002/jcd.21298. [32] C. Wang, Y. Chang and T. Feng, Optimal multiply constant-weight codes from generalized Howell designs, Graphs Combin., 35 (2019), 611-632.  doi: 10.1007/s00373-019-02020-7. [33] C. Wang and B. Du, Existence of generalized Howell designs of side $n+ 1$, Util. Math., 80 (2009), 143-159. [34] X. Wang, H. Wei, C. Shangguan and G. Ge, New bounds and constructions for multiply constant-weight codes, IEEE Trans. Inf. Theory, 62 (2016), 6315-6327.  doi: 10.1109/TIT.2016.2609389. [35] J. Yao, Y. Hu and J. Wang, Existence of generalized Howell designs GHD$(n+5, 3n)$, J. Guangxi Norm. Uni. (Natural Science Edition), 39 (2021), 103-114.
Values $n\geq 7$ for which no GHD $(n+\frac{n-1}{2}, 3n)$ is known
 7 47 51 55 59 63 77 83 87 95 119
 7 47 51 55 59 63 77 83 87 95 119
Values $n\geq 6$ for which no GHD $(n+\frac{n-2}{2}, 3n)$ is known
 48 50 52 56 58 60 62 64 68 72 74 76 78 80 82 84 88 90 92 98 100 104 106 108 110 112 114 116 120 124 126 128 130 132 136 138 142 144 146 152 154 156 158 160 168 170 172 176 180 186 188 190 192 194 196 202 204 208 210 212 218 220 222 224
 48 50 52 56 58 60 62 64 68 72 74 76 78 80 82 84 88 90 92 98 100 104 106 108 110 112 114 116 120 124 126 128 130 132 136 138 142 144 146 152 154 156 158 160 168 170 172 176 180 186 188 190 192 194 196 202 204 208 210 212 218 220 222 224
Values $n\equiv 0\ (\bmod\ 4)$ for which no GHD $(n+\frac{n-4}{2}, 3n)$ is known
 56 64 72 80 96 112 120 136 144 152 176 184 200 220 224
 56 64 72 80 96 112 120 136 144 152 176 184 200 220 224
Values $n\geq 6$ for which no GHD $(n+\frac{n-6}{2}, 3n)$ is known
 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 74 76 78 80 84 88 90 94 96 104 106 108 118 120
 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 74 76 78 80 84 88 90 94 96 104 106 108 118 120
Values $n\geq 7$ for which no GHD $(n+\frac{n-3}{2}, 3n)$ is known
 29 31 33 35 37 41 43 45 47 49 51 53 55 61 63 69 77 95
 29 31 33 35 37 41 43 45 47 49 51 53 55 61 63 69 77 95
Values $n\geq 5$ for which no GHD $(n+\frac{n-5}{2}, 3n)$ is known
 29 31 33 35 37 41 43 47 49 51 53 57 59 61 63 69 73 77 79 81 83 89 91 93 99 101 107 109 111 113 117 121 127 129 131 133 137 139 143 147 153 157 159 161 169 171 173 177 181 187 189 191 193 197 203 209 211 213 219 221 223 233 237 239 241 243 249 257 259 261 267 269 271 275 277 281 283 287 289 297 299 301 303 307 309 313 315 321 323 329 333 337 341 349 351
 29 31 33 35 37 41 43 47 49 51 53 57 59 61 63 69 73 77 79 81 83 89 91 93 99 101 107 109 111 113 117 121 127 129 131 133 137 139 143 147 153 157 159 161 169 171 173 177 181 187 189 191 193 197 203 209 211 213 219 221 223 233 237 239 241 243 249 257 259 261 267 269 271 275 277 281 283 287 289 297 299 301 303 307 309 313 315 321 323 329 333 337 341 349 351
Starters and adders for GHD $(n+3, 3n)$s in Lemma 4.9
 R C $7_0 8_1 3_2$ $2_0 6_1 3_2$ A GHD $(12, 27)$ $4_0 6_1 7_2$ $7_0 4_1 0_2$ with an empty $8_0 4_1 6_2$ $5_0 3_1 2_2$ $3 \times 3$ subarray S A S A S A S A S A $0_0 1_0 3_0$ 0 $0_1 1_1 3_1$ 8 $1_2 2_2 8_2$ 5 $2_0 6_0 5_1$ 2 $7_1 2_1 5_2$ 3 $5_0 0_2 4_2$ 1 R C $8_0 8_1 4_2$ $9_0 0_1 3_2$ A GHD $(13, 30)$ $6_0 2_1 3_2$ $5_0 2_1 0_2$ with an empty $7_0 6_1 8_2$ $8_0 3_1 7_2$ $3 \times 3$ subarray S A S A S A S A S A $0_0 2_0 3_0$ 0 $1_1 4_1 0_1$ 7 $2_2 5_2 1_2$ 4 $5_0 1_0 9_1$ 6 $3_1 5_1 0_2$ 1 $7_2 9_2 9_0$ 5 $4_0 7_1 6_2$ 2 R C $3_0 2_1 6_2$ $10_0 4_1 6_2$ A GHD $(14, 33)$ $1_0 9_1 3_2$ $9_0 1_1 2_2$ with an empty $8_0 3_1 9_2$ $1_0 10_1 9_2$ $3 \times 3$ subarray S A S A S A S A S A $10_0 5_0 6_0$ 9 $1_1 4_1 10_1$ 4 $7_2 10_2 5_2$ 5 $4_0 2_0 6_1$ 3 $7_1 8_1 4_2$ 10 $1_2 2_2 7_0$ 6 $0_0 0_1 0_2$ 0 $9_0 5_1 8_2$ 8
 R C $7_0 8_1 3_2$ $2_0 6_1 3_2$ A GHD $(12, 27)$ $4_0 6_1 7_2$ $7_0 4_1 0_2$ with an empty $8_0 4_1 6_2$ $5_0 3_1 2_2$ $3 \times 3$ subarray S A S A S A S A S A $0_0 1_0 3_0$ 0 $0_1 1_1 3_1$ 8 $1_2 2_2 8_2$ 5 $2_0 6_0 5_1$ 2 $7_1 2_1 5_2$ 3 $5_0 0_2 4_2$ 1 R C $8_0 8_1 4_2$ $9_0 0_1 3_2$ A GHD $(13, 30)$ $6_0 2_1 3_2$ $5_0 2_1 0_2$ with an empty $7_0 6_1 8_2$ $8_0 3_1 7_2$ $3 \times 3$ subarray S A S A S A S A S A $0_0 2_0 3_0$ 0 $1_1 4_1 0_1$ 7 $2_2 5_2 1_2$ 4 $5_0 1_0 9_1$ 6 $3_1 5_1 0_2$ 1 $7_2 9_2 9_0$ 5 $4_0 7_1 6_2$ 2 R C $3_0 2_1 6_2$ $10_0 4_1 6_2$ A GHD $(14, 33)$ $1_0 9_1 3_2$ $9_0 1_1 2_2$ with an empty $8_0 3_1 9_2$ $1_0 10_1 9_2$ $3 \times 3$ subarray S A S A S A S A S A $10_0 5_0 6_0$ 9 $1_1 4_1 10_1$ 4 $7_2 10_2 5_2$ 5 $4_0 2_0 6_1$ 3 $7_1 8_1 4_2$ 10 $1_2 2_2 7_0$ 6 $0_0 0_1 0_2$ 0 $9_0 5_1 8_2$ 8
Starters and adders for GHD $(n+4, 3n)$s in Lemma 4.10
 R C $5_0 5_1 5_2$ $9_0 10_1 2_2$ $4_0 8_1 9_2$ $3_0 5_1 4_2$ A GHD $(15, 33)$ $8_0 6_1 10_2$ $8_0 0_1 6_2$ with an empty $10_0 9_1 6_2$ $2_0 8_1 10_2$ $4 \times 4$ subarray S A S A S A S A S A $2_0 7_0 9_0$ 3 $1_1 2_1 4_1$ 2 $1_2 2_2 4_2$ 7 $0_0 7_1 3_2$ 0 $3_0 6_0 0_1$ 1 $3_1 10_1 8_2$ 10 $1_0 0_2 7_2$ 5 R C $8_0 0_1 1_2$ $11_0 0_1 8_2$ $10_0 6_1 9_2$ $8_0 10_1 4_2$ A GHD $(16, 36)$ $5_0 2_1 6_2$ $1_0 7_1 5_2$ with an empty $9_0 7_1 0_2$ $3_0 2_1 1_2$ $4 \times 4$ subarray S A S A S A S A S A $1_0 4_0 6_0$ 1 $4_1 8_1 9_1$ 7 $2_2 3_2 5_2$ 9 $0_0 5_1 7_2$ 0 $3_0 7_0 10_1$ 3 $1_1 3_1 10_2$ 5 $4_2 8_2 2_0$ 2 $11_0 11_1 11_2$ 10 R C $6_0 12_1 2_2$ $6_0 6_1 11_2$ $2_0 9_1 3_2$ $3_0 4_1 2_2$ A GHD $(17, 39)$ $10_0 6_1 12_2$ $12_0 3_1 5_2$ with an empty $9_0 7_1 7_2$ $10_0 5_1 1_2$ $4 \times 4$ subarray S A S A S A S A S A $4_0 7_0 11_0$ 7 $0_1 1_1 8_1$ 1 $1_2 9_2 10_2$ 12 $3_1 5_1 4_2$ 8 $0_0 8_0 10_1$ 0 $2_1 11_1 6_2$ 10 $5_2 8_2 5_0$ 2 $0_2 11_2 3_0$ 6 $1_0 12_0 4_1$ 3 R C $2_0 3_1 10_2$ $13_0 13_1 0_2$ $1_0 4_1 7_2$ $12_0 5_1 3_2$ An GHD $(18, 42)$ $12_0 2_1 8_2$ $7_0 4_1 9_2$ with an empty $3_0 1_1 12_2$ $3_0 2_1 1_2$ $4 \times 4$ subarray S A S A S A S A S A $4_0 5_0 9_0$ 6 $5_1 0_1 8_1$ 12 $2_2 3_2 11_2$ 10 $11_0 13_0 7_1$ 7 $7_0 10_0 12_1$ 9 $11_1 13_1 13_2$ 11 $1_2 5_2 8_0$ 1 $6_1 10_1 0_2$ 5 $6_2 9_2 6_0$ 2 $0_0 9_1 4_2$ 0
 R C $5_0 5_1 5_2$ $9_0 10_1 2_2$ $4_0 8_1 9_2$ $3_0 5_1 4_2$ A GHD $(15, 33)$ $8_0 6_1 10_2$ $8_0 0_1 6_2$ with an empty $10_0 9_1 6_2$ $2_0 8_1 10_2$ $4 \times 4$ subarray S A S A S A S A S A $2_0 7_0 9_0$ 3 $1_1 2_1 4_1$ 2 $1_2 2_2 4_2$ 7 $0_0 7_1 3_2$ 0 $3_0 6_0 0_1$ 1 $3_1 10_1 8_2$ 10 $1_0 0_2 7_2$ 5 R C $8_0 0_1 1_2$ $11_0 0_1 8_2$ $10_0 6_1 9_2$ $8_0 10_1 4_2$ A GHD $(16, 36)$ $5_0 2_1 6_2$ $1_0 7_1 5_2$ with an empty $9_0 7_1 0_2$ $3_0 2_1 1_2$ $4 \times 4$ subarray S A S A S A S A S A $1_0 4_0 6_0$ 1 $4_1 8_1 9_1$ 7 $2_2 3_2 5_2$ 9 $0_0 5_1 7_2$ 0 $3_0 7_0 10_1$ 3 $1_1 3_1 10_2$ 5 $4_2 8_2 2_0$ 2 $11_0 11_1 11_2$ 10 R C $6_0 12_1 2_2$ $6_0 6_1 11_2$ $2_0 9_1 3_2$ $3_0 4_1 2_2$ A GHD $(17, 39)$ $10_0 6_1 12_2$ $12_0 3_1 5_2$ with an empty $9_0 7_1 7_2$ $10_0 5_1 1_2$ $4 \times 4$ subarray S A S A S A S A S A $4_0 7_0 11_0$ 7 $0_1 1_1 8_1$ 1 $1_2 9_2 10_2$ 12 $3_1 5_1 4_2$ 8 $0_0 8_0 10_1$ 0 $2_1 11_1 6_2$ 10 $5_2 8_2 5_0$ 2 $0_2 11_2 3_0$ 6 $1_0 12_0 4_1$ 3 R C $2_0 3_1 10_2$ $13_0 13_1 0_2$ $1_0 4_1 7_2$ $12_0 5_1 3_2$ An GHD $(18, 42)$ $12_0 2_1 8_2$ $7_0 4_1 9_2$ with an empty $3_0 1_1 12_2$ $3_0 2_1 1_2$ $4 \times 4$ subarray S A S A S A S A S A $4_0 5_0 9_0$ 6 $5_1 0_1 8_1$ 12 $2_2 3_2 11_2$ 10 $11_0 13_0 7_1$ 7 $7_0 10_0 12_1$ 9 $11_1 13_1 13_2$ 11 $1_2 5_2 8_0$ 1 $6_1 10_1 0_2$ 5 $6_2 9_2 6_0$ 2 $0_0 9_1 4_2$ 0
Starter and adder for the GHD $(22, 51)$s in Lemma 4.11
 R C $0_0 0_1 16_2$ $5_0 7_1 3_2$ $13_0 14_1 3_2$ $13_0 0_1 1_2$ A GHD $(22, 51)$ $2_0 10_1 13_2$ $16_0 5_1 0_2$ with an empty $11_0 5_1 2_2$ $10_0 2_1 6_2$ $5 \times 5$ subarray $7_0 3_1 10_2$ $7_0 4_1 13_2$ S A S A S A S A S A $3_0 8_0 13_1$ 0 $11_1 16_1 4_2$ 12 $9_2 15_2 5_0$ 6 $9_0 16_0 1_2$ 10 $9_1 1_1 6_0$ 9 $14_2 6_2 12_1$ 13 $4_0 10_0 12_0$ 2 $4_1 6_1 7_1$ 8 $8_2 11_2 12_2$ 14 $1_0 14_0 15_0$ 3 $2_1 8_1 15_1$ 1 $0_2 5_2 7_2$ 7
 R C $0_0 0_1 16_2$ $5_0 7_1 3_2$ $13_0 14_1 3_2$ $13_0 0_1 1_2$ A GHD $(22, 51)$ $2_0 10_1 13_2$ $16_0 5_1 0_2$ with an empty $11_0 5_1 2_2$ $10_0 2_1 6_2$ $5 \times 5$ subarray $7_0 3_1 10_2$ $7_0 4_1 13_2$ S A S A S A S A S A $3_0 8_0 13_1$ 0 $11_1 16_1 4_2$ 12 $9_2 15_2 5_0$ 6 $9_0 16_0 1_2$ 10 $9_1 1_1 6_0$ 9 $14_2 6_2 12_1$ 13 $4_0 10_0 12_0$ 2 $4_1 6_1 7_1$ 8 $8_2 11_2 12_2$ 14 $1_0 14_0 15_0$ 3 $2_1 8_1 15_1$ 1 $0_2 5_2 7_2$ 7
GHD $(n+3, 3n)$s from 3 HMOLS of type $h^m s^1$ (with $n = hm+s$) in Lemma 4.19
 $n = hm+s$ $h$ $m$ $s$ $n = hm+s$ $h$ $m$ $s$ 45, 50 9, 10 5 0 53-57 9 5 8-12 58-62 10 5 8-12 63-67 11 5 8-12 68-76 12 5 8-16 77-81 9 7 14-18 82-93 9 8 10-21 94-106 12 7 10-22
 $n = hm+s$ $h$ $m$ $s$ $n = hm+s$ $h$ $m$ $s$ 45, 50 9, 10 5 0 53-57 9 5 8-12 58-62 10 5 8-12 63-67 11 5 8-12 68-76 12 5 8-16 77-81 9 7 14-18 82-93 9 8 10-21 94-106 12 7 10-22
GHD $(n+4, 3n)$s from 3 HMOLS of type $h^m s^1$ (with $n = hm+s$) in Lemma 4.22
 $n = hm+s$ $h$ $m$ $s$ $n = hm+s$ $h$ $m$ $s$ 55 11 5 0 64-67 11 5 9-12 69-76 12 5 9-16 77-81 13 5 12-16 82-86 14 5 12-16 87-95 11 7 10-18 96-108 12 7 12-24 109-116 12 8 13-20
 $n = hm+s$ $h$ $m$ $s$ $n = hm+s$ $h$ $m$ $s$ 55 11 5 0 64-67 11 5 9-12 69-76 12 5 9-16 77-81 13 5 12-16 82-86 14 5 12-16 87-95 11 7 10-18 96-108 12 7 12-24 109-116 12 8 13-20
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