Abstract
The study of \((n\times m,k,\lambda _a,\lambda _c)\)-geometric orthogonal codes (\((n\times m,k,\lambda _a,\lambda _c)\)-GOCs) is motivated by the application in DNA origami. The central research on GOCs is to determine the value of \(\Phi (n\times m,k,\lambda _a,\lambda _c)\), i.e., the largest possible size among all \((n\times m,k,\lambda _a,\lambda _c)\)-GOCs. When \(\lambda _a=\lambda _c\), the exact values of \(\Phi (n\times m,3,1,1)\) and \(\Phi (n\times m,k,k-1,k-1)\) were recently determined by Wang and Su et al. In this paper, we research on the cases of \(\lambda _c=k-1\) and \(\lambda _a \le k-2\). We determine the exact values of \(\Phi (n\times m,k,k-2,k-1)\) and \(\Phi (n\times m,k,k-3,k-1)\), and give a calculation method of \(\Phi (n\times m,k,\lambda _a,k-1)\) with \(\lambda _a\le k-4\) for any positive integers n, m and k.
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References
Brualdi R.A.: Introductory Combinatorics. Pearson Education India, India (1977).
Chang Y., Wang X.: Determination of the exact value for \(\Psi (m, k, k-1)\). IEEE Trans. Inf. Theory 57(6), 3810–3814 (2011).
Chee Y.M., Kiah H.M., Ling S., Wei H.: Geometric orthogonal codes of size larger than optical orthogonal codes. IEEE Trans. Inf. Theory 64(4), 2883–2895 (2018).
Chen J., Ji L., Li Y.: New optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4. Des. Codes Cryptogr. 85(2), 299–318 (2017).
Chung F.R.K., Salehi J.A., Wei V.K.: Optical orthogonal codes: design, analysis, and applications. IEEE Trans. Inf. Theory 35, 595–604 (1989).
Doty D., Winslow A.: Design of geometric molecular bonds. IEEE Trans. Mol. Biol. Multi-Scale Commun. 3(1), 13–23 (2017).
Fang Z., Zhou J.: The sizes of maximal \((v, k, k-2, k-1)\) optical orthogonal codes. Des. Codes Cryptogr. 88, 807–824 (2020).
Feng T., Wang L., Wang X.: Optimal 2-D \((n\times m,3,2,1)\)-optical orthogonal codes and related equi-difference conflict avoiding codes. Des. Codes Cryptogr. 87, 1499–1520 (2019).
Gerling T., Wagenbauer K.F., Neuner A.M., Dietz H.: Dynamic DNA devices and assemblies formed by shape-complementary, non-base pairing 3D components. Science 347(6229), 1446–1452 (2015).
Huang Y., Chang Y.: Two classes of optimal two-dimensional OOCs. Des. Codes Cryptogr. 63, 357–363 (2012).
Huang Y., Chang Y.: The sizes of optimal \((n,4,\lambda ,3)\) optical orthogonal codes. Discret. Math. 312, 3128–3139 (2012).
Pan R., Feng T., Wang L., Wang X.: Optimal optical orthogonal signature pattern codes with weight three and cross-correlation constraint one. Des. Codes Cryptogr. 88, 119–131 (2020).
Pan R., Chang Y.: Determination of the sizes of optimal \((m, n, k, \lambda , k-1)\)-OOSPCs for \(\lambda =k-1, k\). Discret. Math. 313, 1327–1337 (2013).
Rothemund P.W.K.: Using lateral capillary forces to compute by self-assembly. Nature 440, 297–302 (2006).
Su X., Wang L., Tian Z.: New bound and constructions for geometric orthogonal codes and geometric 180-rotating orthogonal codes. Adv. Math. Commun. 16, 961–983 (2022).
Wang L., Cai L., Feng T., Tian Z., Wang X.: Geometric orthogonal codes and geometrical difference packings. Des. Codes Cryptogr. 90(8), 1857–1879 (2022).
Wang X., Chang Y.: Further results on optimal \((v,4,2,1)\)-OOCs. Discret. Math. 312, 331–340 (2012).
Woo S., Rothemund P.W.K.: Programmable molecular recognition based on the geometry of DNA nanostructures. Nat. Chem. 3(8), 620–627 (2011).
Yang Y.: New enumeration results about the optical orthogonal codes. Inf. Process. Lett. 40, 85–87 (1991).
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Appendices
Appendix
Proof of Lemma 3.14
For any \(B\in \Gamma _{1,2}\),
where \(2\le i\le k\).
Next, we calculate the exact value of \(|\Gamma _{1,2}|\) by Lemma 3.11. For \(k\ge 5,\) we only need to calculate the number of all parameters \((i,j,\mu ,\nu ,y_1,x_i,y_i)\) corresponding to \(B\in \Gamma _{1,2}\). For \(k=4\), we need further minus half of the number of these parameters under the case of \((i,j)=(3,2)\). We count the number by classifying the different (i, j)s. Under the case of each fixed (i, j), it is easy to get that the number of parameters \((\mu ,x_i)\) is \(\lfloor \frac{n-1}{k-2}\rfloor .\)
(i) When \(\nu >0\), according to the size of \(y_1\) and \(y_i\), we divide it into the following two cases:
(a) If \(y_1=0\le y_i\), similar to the case \(\Gamma _{1,1}\), there also require \(y_i\ne y_{i-1}\) for \(B\in \Gamma _{1,2}\). For each fixed \((i,j=i-1)\), the number of parameters \((\nu ,y_1=0,y_i)\) is \(\sum _{\nu =1}^{M_1}(m-(k-i)\nu )-\lfloor \frac{m-1}{k-2}\rfloor ,\) where \(M_1=\lfloor \frac{m-1}{i-2}\rfloor \) if \(\lfloor \frac{k+2}{2}\rfloor +1\le i\le k\) or \(M_1=\lfloor \frac{m-1}{k-i}\rfloor \) if \(2\le i\le \lfloor \frac{k+2}{2}\rfloor \).
(b) If \(y_1>y_i=0\), note that \(y_i\ne y_{i-1}\) always holds in this case. Therefore for each fixed \((i,j=i-1)\), by the results of the case \(\Gamma _{1,1}\), the number of parameters \((\nu ,y_1,y_i=0)\) is \(\sum _{\nu =1}^{M_2}(m-1-(i-2)\nu ),\) where \(M_2=\min \{\lfloor \frac{m-2}{i-2}\rfloor ,\lfloor \frac{m-1}{k-i}\rfloor \}\) if \(2<i<k\) or \(M_2=\lfloor \frac{m-1}{k-i}\rfloor \) if \(i=2\) or \(M_2=\lfloor \frac{m-2}{i-2}\rfloor \) if \(i=k\).
Combining the above results, when \(k\ge 5\) we have
(ii) When \(\nu =0\), similar to case (i), we get
(When \(k=4\), if we limit \(\nu =\nu '=0\), then there is no intersection sets in \(\Gamma _{1,2}\) by Lemma 3.11. Therefore the above formula for \(|\Gamma ^2_{1,2}|\) also holds for \(k=4\).)
(iii) When \(\nu <0\), by Lemma 3.12, \(|\Gamma ^3_{1,2}|=|\Gamma ^1_{1,2}|\).
When \(k\ge 5\), we get the conclusion immediately by summing the above three cases. When \(k=4\), it is not difficult to calculate that the number of intersection sets is \(\lfloor \frac{n-1}{2}\rfloor \cdot (m^2-m-\lfloor \frac{m-1}{2}\rfloor )\) by Lemma 3.11. Here, we need further remove these intersection sets to get the conclusion after detailed simplification. \(\square \)
Proof of Lemma 3.15
For any \(B\in \Gamma _{2,1}\),
where \(2\le i\le k, 1\le j\le k-1, j\ge i, j-i\equiv 1\pmod {2}\), \((i-2)\mu<x_i<(i-1)\mu \). In particular, when \(k\equiv 0\pmod {2}\) and \((i,j)=(2,k-1)\), \((x_2,y_2)\ne (\frac{\mu }{2},y_1+\frac{\nu }{2})\).
Next, we calculate the exact value of \(|\Gamma _{2,1}|\) by Lemma 3.10. For \(k\ge 4\), we only need to calculate the number of all parameters \((i,j,\mu ,\nu ,y_1,x_i,y_i)\) corresponding to \(B\in \Gamma _{2,1}\). We count the number by classifying the different (i, j)s. Under the case of each fixed (i, j), we first calculate the number of sets satisfying \(\lambda _{\Delta B}((\mu ,\nu ))=k-2\), then remove the subcase of \(\lambda (\Delta B)=k-1\) by Lemma 3.2.
By Lemma 3.5, the number of parameters \((\mu ,x_i)\) is \(\sum _{\mu =1}^{\min \{N'_2,N_2\}}(\mu -1)+\sum _{\mu =N'_2+1}^{N_2}(n-1-\frac{2k-j+i-5}{2}\mu )\), where \(N_2=\lfloor \frac{2n-4}{2k-j+i-5}\rfloor \), \(N'_2=\lfloor \frac{2n}{2k-j+i-3}\rfloor \).
(i) When \(\nu >0\), according to the size of \(y_1\) and \(y_i\), we divide it into the following two cases:
(a) If \(y_1=0\le y_i\), then the conditions \(y_i+\frac{2k-j-i-1}{2}\nu \le m-1\) and \(\frac{j+i-3}{2}\nu \le m-1\) yield \(1\le \nu \le \min \{\lfloor \frac{2(m-1)}{2k-j-i-1}\rfloor ,\lfloor \frac{2(m-1)}{j+i-3}\rfloor \}\triangleq M_3, 0\le y_i\le m-1-\frac{2k-j-i-1}{2}\nu \). Therefore for each fixed (i, j), the number of parameters \((\nu ,y_1=0,y_i)\) is \(\sum _{\nu =1}^{M_3}(m-\frac{2k-j-i-1}{2}\nu ).\)
(b) If \(y_1>y_i=0\), then the conditions \(y_1\ge 1,y_1+\frac{j+i-3}{2}\nu \le m-1\) and \(\frac{2k-j-i-1}{2}\nu \le m-1\) yield \(1\le \nu \le \min \{\lfloor \frac{2(m-1)}{2k-j-i-1}\rfloor ,\lfloor \frac{2(m-2)}{j+i-3}\rfloor \}\triangleq M_4\), \(1\le y_1\le m-1-\frac{j+i-3}{2}\nu \). Therefore for each fixed (i, j), the number of parameters \((\nu ,y_1,y_i=0)\) is \(\sum _{\nu =1}^{M_4}(m-1-\frac{j+i-3}{2}\nu ).\)
For \(k\equiv 0 ~(\textrm{mod}~2)\), (i, j) can take \((2,k-1)\). We further remove the sets satisfying \(\lambda (\Delta B)=k-1\), which holds only when \((i,j)=(2,k-1)\), \((x_2,y_2)=(\frac{\mu }{2},y_1+\frac{\nu }{2})\) and \(y_2>y_1=0\). These sets have the following form:
Therefore, we get
where \(\gamma _k=1\) for \(k\equiv 0 ~(\textrm{mod}~2); \gamma _k=0\) for \(k\equiv 1 ~(\textrm{mod}~2)\).
(ii) When \(\nu =0\), similar to case (i), we can get:
\(~~~~~~~~~~~~~~~~~~-\lfloor \frac{n-1}{k-1}\rfloor \cdot \gamma _k.\)
(iii) When \(\nu <0\), by Lemma 3.12, \(|\Gamma ^3_{2,1}|=|\Gamma ^1_{2,1}|\).
Summing the above three cases, we get the conclusion immediately. \(\square \)
Proof of Lemma 3.16
For \(B\in \Gamma _{2,2}\),
where \(2\le i\le k, 1\le j\le k-1, j\ge i, j-i\equiv 1\pmod {2}\).
Next, we calculate the exact value of \(|\Gamma _{2,2}|\) by Lemma 3.11. For \(k\ge 4\), we first calculate the number of all parameters \((i,j,\mu ,\nu ,y_1,x_i,y_i)\) corresponding to \(B\in \Gamma _{2,2}\), then minus half of the number of parameters under the case of \((i,j)=(2,k-1)\) for \(k\equiv 0\pmod {2}\). We count the number by classifying the different (i, j)s. Under the case of each fixed (i, j), it is easy to know that the number of parameters \((\mu ,x_i)\) is \(\lfloor \frac{2(n-1)}{2k-j+i-5}\rfloor .\)
(i) When \(\nu >0\), according to the size of \(y_1\) and \(y_i\), we divide it into the following two cases:
(a) If \(y_1=0\le y_i\), similar to the case \(\Gamma _{2,1}\), there also require \(y_i\ne y_{i-1}\) in \(\Gamma _{2,2}\). So it’s easy to get for each fixed (i, j), the number of parameters \((\nu ,y_1=0,y_i)\) is \(\sum _{\nu =1}^{M_3}(m-\frac{2k-j-i-1}{2}\nu )-\lfloor \frac{2(m-1)}{2k+i-j-5}\rfloor ,\) where \(M_3=\min \{\lfloor \frac{2(m-1)}{2k-j-i-1}\rfloor ,\lfloor \frac{2(m-1)}{j+i-3}\rfloor \}\).
(b) If \(y_1>y_i=0\), note that \(y_i\ne y_{i-1}\) always holds in this case. Therefore for each fixed (i, j), by the results of the case \(\Gamma _{2,1}\), the number of parameters \((\nu ,y_1,y_i=0)\) is \(\sum _{\nu =1}^{M_4}(m-1-\frac{j+i-3}{2}\nu ),\) where \(M_4=\min \{\lfloor \frac{2(m-1)}{2k-j-i-1}\rfloor ,\lfloor \frac{2(m-2)}{j+i-3}\rfloor \}\).
By Lemma 3.11, when \(k\equiv 0 ~(\textrm{mod}~ 2)\) and \((i,j)=(2,k-1)\), we need to remove the following intersection sets:
and the number of such sets is \(\sum _{\nu =1}^{\lfloor \frac{2(m-2)}{k-2}\rfloor }(m-1-\frac{k-2}{2}\nu )\cdot \lfloor \frac{2(n-1)}{k-2}\rfloor \). Therefore, we get
where \(\gamma _k=1\) for \(k\equiv 0 ~(\textrm{mod}~2); \gamma _k=0\) for \(k\equiv 1 ~(\textrm{mod}~2)\).
(ii) When \(\nu =0\), similar to case (i), we can get
(iii) When \(\nu <0\), \(|\Gamma ^3_{2,2}|=|\Gamma ^1_{2,2}|\).
Summing the above three cases, we get the conclusion immediately. \(\square \)
Proof of Lemma 3.17
For \(B\in \Gamma _{3,1}\),
where \(2\le i\le k, 1\le j\le k-1, j\ge i, j-i\equiv 0\pmod {2}\), \((i-2)\mu<x_i<(i-1)\mu \). In particular, when \(k\equiv 1\pmod {2}\) and \((i,j)=(2,k-1)\), \((x_2,y_2)\ne (\frac{\mu }{2},y_1+\frac{\nu }{2})\).
Next, we calculate the exact value of \(|\Gamma _{3,1}|\) by Lemma 3.10. For \(k\ge 4\), we only need to calculate the number of all parameters \((i,j,\mu ,\nu ,y_1,x_i,y_i)\) corresponding to \(B\in \Gamma _{3,1}\). We count the number by classifying the different (i, j)s. Under the case of each fixed (i, j), we first calculate the number of the sets satisfying \(\lambda _{\Delta B}((\mu ,\nu ))=k-2\), then remove the subcase of \(\lambda (\Delta B)=k-1\) by Lemma 3.2.
By Lemma 3.5, we get the number of parameters \((\mu ,x_i)\) is \(\sum _{\mu =1}^{N_3}(\mu -1),~\textrm{where}~ N_3=\lfloor \frac{2n-2}{2k-j+i-4}\rfloor .\)
(i) When \(\nu >0\), according to the size of \(y_1\) and \(y_i\), we divide it into the following two cases:
(a) If \(y_1=0\le y_i\), then the conditions \(y_i+\frac{j-i}{2}\nu \le m-1\) and \(\frac{2k-j+i-4}{2}\nu \le m-1\) yield \(1\le \nu \le \lfloor \frac{2(m-1)}{2k-j+i-4}\rfloor \triangleq M_5, 0\le y_i\le m-1-\frac{j-i}{2}\nu \). Under the case of each fixed (i, j), the number of parameters \((\nu ,y_1=0,y_i)\) is \(\sum _{\nu =1}^{M_5}(m-\frac{j-i}{2}\nu ). \)
(b) If \(y_1>y_i=0\), then the conditions \(y_1\ge 1,y_1+\frac{2k-j+i-4}{2}\nu \le m-1\) and \(\frac{j-i}{2}\nu \le m-1\) yield \(1\le \nu \le M_6\), where \(M_6=\min \{\lfloor \frac{2(m-2)}{2k-j+i-4}\rfloor ,\lfloor \frac{2(m-1)}{j-i}\rfloor \}\) if \(i\ne j\) or \(M_6=\lfloor \frac{m-2}{k-2}\rfloor \) if \(i=j\), \(1\le y_1\le m-1-\frac{2k-j+i-4}{2}\nu \). Therefore for each fixed (i, j), the number of parameters \((\nu ,y_1,y_i=0)\) is \(\sum _{\nu =1}^{M_6}(m-1-\frac{2k-j+i-4}{2}\nu ).\)
For \(k\equiv 1 ~(\textrm{mod}~2)\), (i, j) can take \((2,k-1)\). Similar to \(\Gamma _{2,1}\), we need to remove the following sets:
Therefore, we get
\(~~~~~~~~~~~~~~~~-\lfloor \frac{n-1}{k-1}\rfloor \cdot \lfloor \frac{m-1}{k-1}\rfloor \cdot (1-\gamma _k),\)
where \(\gamma _k=1\) for \(k\equiv 0 ~(\textrm{mod}~2); \gamma _k=0\) for \(k\equiv 1 ~(\textrm{mod}~2)\).
(ii) When \(\nu =0\), similar to case (1), we can get:
(iii) When \(\nu <0\), by Lemma 3.12, \(|\Gamma ^3_{3,1}|=|\Gamma ^1_{3,1}|\).
Summing the above three cases, we get the conclusion immediately. \(\square \)
Proof of Lemma 3.18
For \(B\in \Gamma _{3,2}\),
where \(2\le i\le k, 1\le j\le k-1, j\ge i, j-i\equiv 0\pmod {2}\).
Next, we calculate the exact value of \(|\Gamma _{3,2}|\) by Lemma 3.11. For \(k\ge 4\), we only need to calculate the number of all parameters \((i,j,\mu ,\nu ,y_1,x_i,y_i)\) corresponding to \(B\in \Gamma _{3,2}\). We count the number by classifying the different (i, j)s. Under the case of each fixed (i, j), we first calculate the number of parameters \((\mu ,x_i)\) is \(\lfloor \frac{2(n-1)}{2k-j+i-4}\rfloor .\)
(i) When \(\nu >0\), according to the size of \(y_1\) and \(y_i\), we divide it into the following two cases:
(a) If \(y_1=0\le y_i\), similar to the case \(\Gamma _{3,1}\), there also require \(y_i\ne y_{i-1}\) in \(\Gamma _{3,2}\). For each fixed (i, j), it’s easy to get that the number of parameters \((\nu ,y_1=0,y_i)\) is \(\sum _{\nu =1}^{M_5}(m-1-\frac{j-i}{2}\nu ),~\textrm{where}~ M_5=\lfloor \frac{2(m-1)}{2k-j+i-4}\rfloor .\)
(b) If \(y_1>y_i=0\), note that \(y_i\ne y_{i-1}\) always holds in this case. Therefore for each fixed (i, j), by the results of the case \(\Gamma _{3,1}\), the number of parameters \((\nu ,y_1,y_i=0)\) is \(\sum _{\nu =1}^{M_6}(m-1-\frac{2k-j+i-4}{2}\nu )\), where \(M_6=\min \{\lfloor \frac{2(m-2)}{2k-j+i-4}\rfloor ,\lfloor \frac{2(m-1)}{j-i}\rfloor \}\) if \(i\ne j\) or \(M_6=\lfloor \frac{m-2}{k-2}\rfloor \) if \(i=j\).
Therefore, we get
(ii) When \(\nu =0\), similar to case (i), we can get
(iii) When \(\nu <0\), by Lemma 3.12, \(|\Gamma ^3_{3,2}|=|\Gamma ^1_{3,2}|\).
Summing the above three cases, we get the conclusion immediately. \(\square \)
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Su, X., Tian, Z. & Hao, G. Determination of the sizes of optimal geometric orthogonal codes with parameters \((n\times m,k,\lambda ,k-1)\). Des. Codes Cryptogr. 92, 365–395 (2024). https://doi.org/10.1007/s10623-023-01312-7
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DOI: https://doi.org/10.1007/s10623-023-01312-7