# Cyclotomic Trace Codes
## 3 Construction of Linear Codes
### Codes Obtained from Fields of Odd Order  
In Reference [7], [PERSON] presented the following construction of linear codes from nonzero squares. Let \(D=\{d_{1},d_{2},\ldots,d_{n}\}\subseteq GF(q)\), such that \(d_{i}\) are nonzero squares in a field \(GF(q)\) and \(q=p^{m}\). Let \(Tr\) denote the trace function defined in the following way. Let \(L=GF(p^{m})\) be a finite extension of a field \(K=GF(p)\). If \(\alpha\) is in \(L\), the trace of \(\alpha\) is the sum  
\[Tr_{L/K}(\alpha)=\alpha+\alpha^{p}+\cdots+\alpha^{p^{m-1}}.\]  
One can define a linear code of length \(n\) over \(GF(p)\) by  
\[C_{D}=\{(Tr(xd_{1}),Tr(xd_{2}),\ldots,Tr(xd_{n})):x\in GF(q)\},\]  
and call \(D\) the defining set of this code \(C_{D}\). Note that by definition, the dimension of the code \(C_{D}\) is at most \(m\). **Result 1** (Corollary 4, [7]). : _Let \(D\) be the set of all quadratic residues in \(GF(p^{m})^{*}\), where \(p\) is an odd prime. If \(m\) is odd, then \(C_{D}\) is a one-weight code over \(GF(p)\) with parameters \([(q-1)/2,m,(p-1)q/2p]\). If \(m\) is even, then \(C_{D}\) is a two-weight code over \(GF(p)\) with parameters \([(q-1)/2,m,(p-1)(q-\sqrt{q})/2p]\) and weight enumerator \(1+\frac{q-1}{2}2^{(p-1)(q-\sqrt{q})/2p}+\frac{q-1}{2}z^{(p-1)(q+\sqrt{q})/2p}\)._  
If \(D\) is a set of nonzero squares in \(GF(q)\), and depending whether \(q\equiv 3\pmod{4}\) or \(q\equiv 1\pmod{4}\), using [PERSON]'s result presented in Reference Corollary 1, we obtain one or two-weight codes. From the supports of all codewords of minimum weight of the one-weight codes, one obtains the family of symmetric designs known as point-hyperplane designs. Since all linear one-weight codes are characterized (see References [10; 11] for more information), in this paper we were interested only in codes with more weights. In Table 1 we describe the two-weight codes obtained by applying the construction of Ding described above, and the corresponding strongly regular graphs obtained from these codes by the well-known construction (see Reference [13]). We give the parameters of the codes and their weight distribution and the parameters of the strongly regular graphs (SRGs). **Remark 1**. : _All strongly regular graphs in Table 1 are Paley graphs._  
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 cubes in a field \(GF(q)\) and \(q=p^{m}\). In Table 2 we describe the codes obtained by applying the construction described above. We give the parameters of the codes, their weight distributions and the descriptions of the SRGs related to constructed codes. **Remark 2**. : _We have constructed SRGs with parameters \(v=q,k=(q-1)/3,\lambda=[(\sqrt{q}+1)^{2}-9]/9,\mu=(\sqrt{q}+1)(\sqrt{q}-2)/9\), and \(q=25,121,289,15625\). According to [PERSON]'s table (see Reference [22]), known graphs with these parameters are obtainable from orthogonal arrays \(OA(\sqrt{q},(\sqrt{q}+1)/3)\). Since our method does not use orthogonal arrays, it is likely that our graphs are new. The SRG with parameters (625,208,63,72) is isomorphic to the SRG obtained from projective 5-ary \([52,4]\) code with weights 40, 45 (see [23]). In fact, the code \([208,4,160]_{5}\) is four copies of the above mentioned projective \([52,4,40]\) code. SRG(14641,4880,1599,1640) is isomorphic to the graph obtained by \(CY2\) construction (see Reference [23]) for \(q=11\) 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 4 in a field \(GF(q)\) and \(q=p^{m}\). The results are given in Table 3.