# Cyclotomic Trace Codes
## 3 Construction of Linear Codes
### Codes Obtained from Fields of Odd Order
\begin{table}
\begin{tabular}{c c c c c} \hline \(p,q,m\) & **Code** & **Weight Dist.** & **Corr. Proj. Code** & **SRG** \\ \hline \(3,9,2\) & \([4,2,2]_{3}\) & \(<0,1>,<2,4>,<4,4>\) & \([2,2,1]^{*}\), \(d^{\perp}=2\) & (9,4,1,2) \\ \(3,81,4\) & \([40,4,24]_{3}\) & \(<0,1>,<24,40>,<30,40>\) & \([20,4,12]^{*}\), \(d^{\perp}=3\) & (81,40,19,20) \\ \(3,729,6\) & \([364,6,234]_{3}\) & \(<0,1>,<234,364>,<252,364>\) & \([182,6,117]\), \(d^{\perp}=3\) & (729,364,181,182) \\ \hline \(5,25,2\) & \([12,2,8]_{5}\) & \(<0,1>,<8,12>,<12,12>\) & \([3,2,2]^{*}\), \(d^{\perp}=3\) & (25,12,5,6) \\ \(5,625,4\) & \([312,4,240]_{5}\) & \(<0,1>,<240,312>,<260,312>\) & \([78,4,60]\), \(d^{\perp}=3\) & (625,312,155,156) \\ \hline \(7,\,49,2\) & \([24,2,18]_{7}\) & \(<0,1>,<18,24>,<24,24>\) & \([4,2,3]^{*}\), \(d^{\perp}=3\) & (49,24,11,12) \\ \(7,\,2401,4\) & \([1200,4,1008]_{7}\) & \(<0,1>,<1008,1200>,<1050,1200>\) & \([200,4,168]\), \(d^{\perp}=3\) & (2401,1200,599,600) \\ \hline \end{tabular}
\end{table} 
Table 1: Two-weight trace codes from nonzero squares. \begin{table}
\begin{tabular}{c c c c c} \hline \(p,q,m\) & **Code** & **Weight Dist.** & **Corr. Proj. Code** & **SRG** \\ \hline \(5,25,2\) & \([8,2,4]_{5}\) & \(<0,1>,<4,8>,<8,16>\) & \([2,2,1]^{*}\), \(d^{\perp}=2\) & (25,8,3,2) \\ \(5,625,4\) & \([208,4,160]_{5}\) & \(<0,1>,<160,416>,<180,208>\) & \([52,4,40]^{*}\), \(d^{\perp}=3\) & (625,208,63,72) \\ \(5,15625,6\) & \([5208,6,4100]_{5}\) & \(<0,1>,<4100,5208>,<4200,10416>\) & \([1302,6,1025]\), \(d^{\perp}=3\) & (15625,5208,1763,1722) \\ \hline \(11,\,121,2\) & \([40,2,30]_{11}\) & \(<0,1>,<30,40>,<40,80>\) & \([4,2,3]^{*}\), \(d^{\perp}=3\) & (121,40,15,12) \\ \(11,\,14641,4\) & \([4880,4,4400]_{11}\) & \(<0,1>,<4400,9760>,<4510,4880>\) & \([488,4,440]\), \(d^{\perp}=3\) & (14641,4880,1599,1640) \\ \(17,\,289,2\) & \([96,2,80]_{17}\) & \(<0,1>,<80,96>,<96,192>\) & \([6,2,5]^{*}\), \(d^{\perp}=3\) & (289,96,35,30) \\ \hline \end{tabular}
\end{table} 
Table 2: Two-weight trace codes from nonzero cubes. **Remark 3**. : _SRG(81,20,1,6) and SRG(49,12,5,2) are unique SRGs with these parameters. The SRG with parameters (81,20,1,6) is isomorphic to the SRG obtained from projective ternary \([10,4]\) code with weights 6, 9 (see Reference [24]). In fact, the code \([20,4,12]_{3}\) is two copies of the above-mentioned projective \([10,4,6]\) code. SRGs with parameters (25,12,5,6), (15625,7812,3905,3906), (169,84,41,42) and (289,144,71,72) are Paley graphs. We have constructed SRGs with parameters (121,30,11,6), (361,90,29,20) and (14641,3660,869,871). According to [PERSON]'s table (see Reference [22]), known graphs with these parameters are obtainable from orthogonal arrays. The graph with parameters (729,182,55,42) does not arise from orthogonal array, although it has suitable parameters. It is isomorphic with the SRG obtained from projective ternary \([91,6]\) code with weights 54, 63 (see Reference [22]). The code \([182,6,108]_{3}\) is two copies of the above mentioned projective \([91,6,54]\) code. The SRG(2401,600,131,156) is isomorphic to the graph obtained by CY2 construction (see Reference [23]), for \(q=7\) and \(S=4\)._  
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 5 in a field \(GF(q)\) and \(q=p^{m}\). The results are given in Table 4.  
**Remark 4**. : _The SRG(81,16,7,2) is a unique strongly regular graph with these parameters. The SRG with parameters (81,16,7,2) is isomorphic to the SRG obtained from the projective ternary [8,4] code with weights 3, 6 (see Reference [13]). The code \([16,4,6]_{3}\) is two copies of the above-mentioned projective \([8,4,3]\) code. Further, we have constructed an SRG with parameters (361,72,23,12), and according to [PERSON]'s table (see Reference [22]), a known graph with these parameters is obtainable from the orthogonal array.