# Cyclotomic Trace Codes
## 4 Support Designs and Generalized Block Graphs
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\boldsymbol{p,q,m}\) & \(\boldsymbol{s}\) & **Code** & **Weight Dist.** & **Support Design** \\ \hline
3,81,4 & 4 & \([10,4,6]_{3}\) & \(<0,1>,<6,60>,<9,20>\) & 3-(10,4,1), \(b=30\) \\
5,625,4 & 6 & \([26,4,20]_{5}\) & \(<0,1>,<20,520>,<25,100>\) & 3-(26,6,1), \(b=130\) \\
4,64,3 & 9 & \([7,3,4]_{4}\) & \(<0,1>,<4,21>,<6,42>\) & 2-(7,3,1), \(b=7\) \\
4,256,4 & 5 & \([17,4,12]_{4}\) & \(<0,1>,<12,204>,<16,51>\) & 3-(17,5,1), \(b=68\) \\
4,1024,5 & 11 & \([31,5,16]_{4}\) & \(<0,1>,<16,93>,<24,930>\) & 2-(31,15,7), \(b=31\) \\
4,1024,5 & 11 & \([31,5,16]_{4}\) & \(<0,1>,<16,93>,<24,930>\) & 2-(31,7,7), \(b=155\) \\
4,1024,5 & 31 & \([11,5,6]_{4}\) & \(<0,1>,<6,155>,<7,165>,<8,165>\) & 2-(11,5,10), \(b=55\) \\
4,1024,5 & 31 & \([11,5,6]_{4}\) & \(<0,1>,<6,155>,<7,165>,<8,165>\) & 2-(11,3,3), \(b=55\) \\  & & & \(<9,330>,<10,165>,<11,33>\) & \\
4,1024,5 & 31 & \([11,5,6]_{4}\) & \(<0,1>,<6,155>,<7,165>,<8,165>\) & 2-(11,4,6), \(b=55\) \\  & & & & \(<9,330>,<10,165>,<11,33>\) & \\
4,16384,7 & 129 & \([127,7,64]_{4}\) & \(<0,1>,<64,381>,<96,16002>\) & 2-(127,63,31), \(b=127\) \\
4,16384,7 & 129 & \([127,7,64]_{4}\) & \(<0,1>,<16,93>,<24,930>\) & 2-(127,31,155), \(b=2667\) \\
8,4096,4 & 45 & \([13,4,9]_{8}\) & \(<0,1>,<9,364>,<10,546>\) & 2-(13,4,4), \(b=52\) \\  & & & & \(<11,1092>,<12,1365>,<13,728>\) & \\
8,4096,4 & 45 & \([13,4,9]_{8}\) & \(<0,1>,<9,364>,<10,546>\) & 2-(13,3,3), \(b=78\) \\  & & & & \(<11,1092>,<12,1365>,<13,728>\) & \\
8,4096,6 & 9 & \([65,4,56]_{8}\) & \(<0,1>,<56,3640>,<64,455>\) & 3-(65,9,1), \(b=520\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Support designs. a generalization of the construction of the block graph from a quasi-symmetric design, so we call a graph constructed in that way a generalized block graph. In Table 10 we list distance regular graphs, including strongly regular graphs, constructed from support designs of the codes constructed by applying the construction described in Section 3. **Remark 10**. : _SRG(10,3,0,1) is the unique graph with these parameters, known as the Petersen graph. SRG(16,6,2,2) is isomorphic to the graph described in Reference [22], constructed from projective \([6,4,2]_{2}\) code with weights 2, 4. SRGs with parameters (55,18,9,4) and (78,22,11,4) are triangular graphs \(T(11)\) and \(T(13)\), respectively. The strongly regular graph (169,36,13,6) has parameters that correspond to the SRG that can be obtained from orthogonal array \(OA(13,3)\). The graphs with parameters (155,42,17,9) and (2667,186,65,9) are block graphs of the quasi-symmetric designs constructed in Table 9. The distance regular graph on 30 vertices of diameter 4 is the unique graph with this intersection array and is called Tutte's 8-cage. Further, the distance regular graph on 68 vertices of diameter 3 is the unique graph with this intersection array and is called Doro graph. For more information we refer the reader to Reference [21]._  
This is a joint collaboration with all three authors contributing substantially throughout. [PERSON] and [PERSON] were funded by Croatian Science Foundation under the project 6732. We thank the reviewers for carefully reading the manuscript and making several useful remarks. The authors declare no conflict of interest.