# Cyclotomic Trace Codes
## 3 Construction of Linear Codes
### Codes Obtained from Fields of Even Order
: _SRGs with parameters (64,21,8,6), (256,51,2,12) and (1024,93,32,6) are isomorphic to the graphs described in Reference [22]. The graph with parameters \((4096,819,194,156)\) is isomorphic to the graph obtained by CY1 construction (see Reference [23]). The graph with parameters \((4096,1365,440,462)\) is isomorphic to the graph obtained by CY4 construction, although corresponding projective codes are not isomorphic and cannot be obtained by CY4 construction. The graph with parameters \((4096,315,74,20)\) is isomorphic to the graph obtained by CY4 construction. The strongly regular graph \((16384,381,128,6)\) has parameters that correspond to the SRG that can be obtained from a orthogonal array \(OA(128,3)\)._  
The codes described in Table 7 are a good source for obtaining quantum codes (see References ([25; 26; 27])). The case where \(q=4\) is motivated by the construction of quantum-error-correcting codes from Hermitian self-orthogonal linear \(GF(4)\) codes. In particular, given a Hermitian self-orthogonal \([n,k]_{4}\) code \(C\) such that no codeword in \(C^{\perp}\setminus C\) has a weight less than \(d\), one can construct a quantum \([n,n-2k,d]\) code, (see Reference [25] Theorems 2, 3). Projective codes from Table 7 give rise to quantum codes over \(GF(4)\). According to the Grassl table [17], these quantum codes are optimal or near optimal. Our construction is different from the constructions given in Reference [17], so our codes are probably new. 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 \(GF(8)\) and \(q=8^{m}\) such that \(s|(8^{m}-1)\). **Remark 8**. : _The SRGs described in Table 8 are isomorphic to the SRGs described in Reference [23] by CY1 and CY4 constructions--the SRGs with parameters (4096,1365,440,462) and (4096,455,6,56) by CY1 construction, and the SRGs with parameters (4096,819,194,156) and (4096,315,74,20) by CY4 construction._ \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(p,q,m\) & \(s\) & **Code** & **Weight Dist.** & **Corr. Proj. Code** & **SRG** \\ \hline
4,64, 3 & 3 & \([21,3,12]_{4}\) & \(<0,1>,<12,21>,<18,42>\) & \([7,3,4]^{\star}\), \(d^{\perp}=3\) & (64,21,8,6) \\
4,64, 3 & 9 & \([7,3,4]_{4}^{\star}\) & \(<0,1>,<4,21>,<6,42>,d^{\perp}=3\) & & (64,21,8,6) \\ \hline
4,256, 4 & 5 & \([51,4,36]_{4}\) & \(<0,1>,<36,204>,<48,51>\) & \([17,4,12]^{\star}\), \(d^{\perp}=4\) & (256,51,2,12) \\
4,256, 4 & 15 & \([17,4,12]_{4}^{\star}\) & \(<0,1>,<12,204>,<6,51>,d^{\perp}=4\) & & (256,51,2,12) \\ \hline
4,1024, 5 & 11 & \([93,5,48]_{4}\) & \(<0,1>,<48,93>,<72,930>\) & \([31,5,16]_{4}\) & \(d^{\perp}=3\) & (1024,93,32,6) \\
4,1024, 5 & 33 & \([31,5,16]_{4}\) & \(<0,1>,<16,93>,<24,930>,d^{\perp}=3\) & & (1024,93,32,6) \\ \hline
4,4096, 6 & 3 & \([1365,6,1008]_{4}\) & \(<0,1>,<1008,2730>,<1056,1365>\) & \([455,6,336]_{4}\) & \(d^{\perp}=3\) & (4096,1365,440,462) \\
4,4096, 6 & 5 & \([819,6,576]_{4}\) & \(<0,1>,<576,819>,<624,3276>\) & \([273,6,192]_{4}^{\perp}=3\) & (4096,819,194,156) \\
4,4096, 6 & 9 & \([455,6,336]_{4}\) & \(<0,1>,<136,2730>,<352,1365>,d^{\perp}=3\) & & (4096,1365,440,462) \\
4,4096, 6 & 13 & \([315,6,192]_{4}\) & \(<0,1>,<192,315>,<240,3780>\) & \([105,6,64],d^{\perp}=3\) & (4096,315,74,20) \\ \hline
4,16384, 7 & 43 & \([381,7,192]_{4}\) & \(<0,1>,<192,381>,<288,16002>\) & \([127,7,64]_{4}\) & \(d^{\perp}=3\) & (16384,381,128,6) \\
4,16384, 7 & 129 & \([127,7,64]_{4}\) & \(<0,1>,<64,381>,<96,16002>,d^{\perp}=3\) & & (16384,381,128,6) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Two-weight quaternary trace codes.