# Cyclotomic Trace Codes
## 3 Construction of Linear Codes
### Codes Obtained from Fields of Even Order  
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 \(s\) in a field \(GF(2)\) and \(q=2^{m}\) such that \(s|(2^{m}-1)\). In Table 6, we describe two-weight projective binary codes. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(p,q,m\) & \(s\) & **Code** & **Weight Dist.** & **SRG** \\ \hline
2,16, 4 & 3,6 & \([5,4,2]_{2}^{*},d^{\perp}=5\) & \(<0,1>,<2,10>,<4,5>\) & (16,5,0,2) \\ \hline
2,64, 6 & 3,6 & \([21,6,8]_{2},d^{\perp}=3\) & \(<0,1>,<8,21>,<12,42>\) & (64,21,8,6) \\ \hline
2,256, 8 & 3,6 & \([85,8,40]_{2}^{*},d^{\perp}=3\) & \(<0,1>,<40,170>,<48,85>\) & (256,85,24,30) \\
2,256, 8 & 5 & \([51,8,24]_{2}^{*},d^{\perp}=3\) & \(<0,1>,<24,204>,<32,51>\) & (256,51,2,12) \\ \hline
2,1024, 10 & 3,6 & \([341,10,160]_{2},d^{\perp}=3\) & \(<0,1>,<160,341>,<176,682>\) & (1024,341,120,110) \\ \hline
2,4096, 12 & 3,6 & \([1365,12,672]_{2},d^{\perp}=3\) & \(<0,1>,<672,2730>,<704,1365>\) & (4096,1365,440,462) \\
2,4096, 12 & 5 & \([819,12,348]_{2},d^{\perp}=3\) & \(<0,1>,<384,819>,<416,3276>\) & (4096,819,194,156) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Two-weight binary trace codes from nonzero powers of 3, 5 and 6. \begin{table}
\begin{tabular}{c c c c c} \hline \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) \\ \(3,729,6\) & \([364,6,236]_{3}\) & \(<0,1>,<234,364>,<252,364>\) & \([182,6,117],d^{\perp}=3\) & (729,364,181,182) \\ \hline \(5,625,4\) & \([104,4,80]_{5}\) & \(<0,1>,<80,520>,<100,104>\) & \([26,4,20]^{*},d^{\perp}=4\) & (625,104,3,20) \\ \(5,15625,6\) & \([2604,6,2000]_{5}\) & \(<0,1>,<2000,2604>,<2100,13020>\) & \([651,6,500],d^{\perp}=3\) & (15625,2604,503,420) \\ \hline \(7,49,2\) & \([8,2,6]_{7}\) & \(<0,1>,<6,24>,<8,24>\) & \([4,2,3]^{*},d^{\perp}=3\) & (49,24,11,12) \\ \(7,2401,4\) & \([400,4,336]_{7}\) & \(<0,1>,<336,1200>,<350,1200>\) & \([200,4,168],d^{\perp}=3\) & (2401,1200,599,600) \\ \hline \(11,121,2\) & \([20,2,10]_{11}\) & \(<0,1>,<10,20>,<20,100>\) & \([2,2,1]^{*},d^{\perp}=2\) & (121,20,9,2) \\ \(11,14641,4\) & \([24404,4,2200]_{11}\) & \(<0,1>,<200,12200>,<2310,2440>\) & \([244,4,220]^{*},d^{\perp}=3\) & (14641,2440,420,341) \\ \hline \(13,169,2\) & \([28,2,24]_{13}\) & \(<0,1>,<24,84>,<28,84>\) & \([7,2,6]^{*},d^{\perp}=3\) & (169,84,41,42) \\ \(13,28561,4\) & \([4760,4,4368]_{13}\) & \(<0,1>,<4368,14280>,<4420,14280>\) & \([1190,4,1092],d^{\perp}=3\) & (28561,14280,7139,7140) \\ \hline \(17,289,2\) & \([48,2,32]_{17}\) & \(<0,1>,<32,48>,<48,240>\) & \([3,2,2]^{*},d^{\perp}=3\) & (289,48,17,6) \\ \hline \(19,361,2\) & \([60,2,54]_{19}\) & \(<0,1>,<54,180>,<60,180>\) & \([10,2,9]^{*},d^{\perp}=3\) & (361,180,89,90) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Two-weight trace codes from powers of 6. **Remark 6**. : _Strongly regular graphs with parameters (16,5,0,2), (64,21,8,6), (256,85,24,30), (1024,341,120,110) and (256,51,2,12) are isomorphic to the graphs described in Reference [22]. The graph with parameters \((4096,1365,440,462)\) is isomorphic to the graph obtained by CY1 construction and the graph \((4096,819,194,156)\) is isomorphic to the graph obtained by CY4 construction (see Reference [23])._  
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 \(s\) in a field \(GF(4)\) and \(q=4^{m}\) such that \(s|(4^{m}-1)\). In Table 7 we describe two-weight quaternary codes. Notice that in these cases, some of the obtained two-weights codes have odd dimension, which was not the case in the previous section. **Remark 7**.