# Cyclotomic Trace Codes
## 3 Construction of Linear Codes
### Codes Obtained from Fields of Odd Order
Moreover, SRG(2401,480,119,90) and SRG(28561,5712,1223,1122) have parameters that correspond to SRGs that can be obtained from orthogonal arrays \(OA(49,10)\) and \(OA(169,34)\), respectively._ \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(p,q,m\) & **Code** & **Weight Dist.** & **Corr. Proj. Code** & **SRG** \\ \hline \(3,81,4\) & \([16,4,6]_{3}\) & \(<0,1>,<6,16>,<12,64>\) & \([8,4,3]\), \(d^{\perp}=3\) & (81,16,7,2) \\ \hline \(7,2401,4\) & \([480,4,378]_{7}\) & \(<0,1>,<378,480>,<420,1920>\) & \([80,4,63]\), \(d^{\perp}=3\) & (2401,480,119,90) \\ \hline \(13,28561,4\) & \([5712,4,5148]_{13}\) & \(<0,1>,<5148,5712>,<5304,22848>\) & \([476,4,429]\), \(d^{\perp}=3\) & (28561,5712,1223,1122) \\ \hline \(19,361,2\) & \([72,2,54]_{19}\) & \(<0,1>,<54,72>,<72,288>\) & \([4,2,3]^{*}\), \(d^{\perp}=3\) & (361,72,23,122) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Two-weight trace codes from powers of 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(p,q,m\) & **Code** & **Weight Dist.** & **Corr. Proj. Code** & **SRG** \\ \hline \(3,81,4\) & \([20,4,12]_{3}^{*}\) & \(<0,1>,<12,60>,<18,20>\) & \([10,4,6]^{*}\), \(d^{\perp}=4\) & (81,20,1,6) \\ \(3,729,6\) & \([182,6,108]_{3}\) & \(<0,1>,<108,182>,<126,546>\) & \([91,6,54]\), \(d^{\perp}=3\) & (729,182,55,42) \\ \hline \(5,25,2\) & \([6,2,4]_{5},d^{\perp}=3\) & \(<0,1>,<4,12>,<6,12>\) & & (25,12,5,6) \\ \(5,15625,6\) & \([3906,6,3100]_{5}^{*}\) & \(<0,1>,<3100,7812>,<3150,7812>\) & \([1953,6,1550]\), \(d^{\perp}=3\) & (15625,7812,3905,3906) \\ \hline \(7,49,2\) & \([12,2,6]_{7}\) & \(<0,1>,<6,12>,<12,36>\) & \([2,2,1]^{*}\), \(d^{\perp}=2\) & (49,12,5,2) \\ \(7,2401,4\) & \([600,4,504]_{7}\) & \(<0,1>,<504,1800>,<546,600>\) & \([100,4,84]^{*}\), \(d^{\perp}=3\) & (2401,600,131,156) \\ \hline \(11,121,2\) & \([30,2,20]_{11}\) & \(<0,1>,<20,30>,<30,90>\) & \([3,2,2]^{*}\), \(d^{\perp}=3\) & (121,30,11,6) \\ \(11,14641,4\) & \([3660,4,330]_{11}\) & \(<0,1>,<3300,10980>,<3410,3660>\) & \([366,4,330]\), \(d^{\perp}=3\) & (14641,3660,869,871) \\ \hline \(13,169,2\) & \([42,2,36]_{13}\) & \(<0,1>,<36,84>,<42,84>\) & \([7,2,6]^{*}\), \(d^{\perp}=3\) & (169,84,41,42) \\ \hline \(17,289,2\) & \([72,2,64]_{17}\) & \(<0,1>,<64,144>,<72,144>\) & \([9,2,8]^{*}\), \(d^{\perp}=3\) & (289,144,71,72) \\ \hline \(19,361,2\) & \([90,2,72]_{19}\) & \(<0,1>,<72,90>,<90,270>\) & \([5,2,4]^{*}\), \(d^{\perp}=3\) & (361,90,29,20) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Two-weight trace codes from powers of 4. Using an approach similar to that described above, we obtained results by defining the set \(D=\{d_{1},d_{2},\ldots,d_{n}\}\subseteq GF(q)\), such that \(d_{i}\) are all nonzero powers of 6 in a field \(GF(q)\) and \(q=p^{m}\). The results are given in Table 5.  
**Remark 5**. : _The SRG(121,20,9,2) is a unique SRG with these parameters. SRGs with parameters (9,4,1,2), (81,40,19,20), (729,364,181,182), (49,24,11,12), (2401,1200,599,600), (169,84,41,42), (28561,14280,7139,7140) and (361,180,89,90) are Paley graphs. Further, we have constructed an SRG with parameters (625,104,3,20), which is isomorphic to the graph described in Reference [22]. Moreover, we have constructed an SRG with parameters (289,48,17,6) and according to [PERSON]'s table (see Reference [22]) a known graph with these parameters is obtainable from orthogonal array. The SRG(15625,2604,503,420) has parameters that correspond to an SRG that can be obtained from an orthogonal array \(OA(125,21)\). The SRG(14641,2440,420,341) is isomorphic to the graph obtained by CY2 construction (see Reference [23]), for \(q=11\) and \(S=4\)._