# Cyclotomic Trace Codes
## 2 Preliminaries  
For background reading in coding theory we refer the reader to Reference [16]. A \(q\)-ary linear code \(C\) of dimension \(k\) for a prime power \(q\), is a \(k\)-dimensional subspace of a vector space \(\mathbb{F}_{q}^{n}\). The elements of \(C\) are called codewords. The Hamming distance between two vectors \(x=(x_{1},...,x_{n}),y=(y_{1},...,y_{n})\in\mathbb{F}_{q}^{n}\) is the number \(\ d(x,y)=|\{i:x_{i}\
eq y_{i}\}|\). The minimum distance \(d\) of a code \(C\) is defined as \(\ d=\min\{d(x,y):x,y\in C,\ x\
eq y\}\). The weight of a codeword \(x\) is \(w(x)=d(x,0)=|\{i:x_{i}\
eq 0\}|\). For a linear code, \(d=\min\{w(x):x\in C,x\
eq 0\}\). A \(q\)-ary linear code of length \(n\), dimension \(k\) and distance \(d\) is called a \([n,k,d]_{q}\) code. Let \(w_{i}\) denote the number of codewords of weight \(i\) in a code \(C\) of length \(n\). The weight distribution of \(C\) is the list \([\langle i,w_{i}\rangle:0\leq i\leq n]\). A one-weight code is a code which has only one nonzero weight and a two-weight code is a code which has only two nonzero weights. A linear code \(C\) is called projective if the minimum distance of its dual code \(C^{\perp}=\{y\ |\ y\cdot x=0\ \text{for all}\ x\in C\}\) is greater than 2. An \([n,k]\) linear code \(C\) is optimal if the minimum weight of \(C\) achieves the theoretical upper bound on the minimum weight of an \([n,k]\) linear code, and near-optimal if its minimum weight is at most 1 less than the largest possible value. An \([n,k]\) linear code \(C\) is said to be a best known linear \([n,k]\) code if \(C\) has the highest minimum weight among all known \([n,k]\) linear codes. A catalog of the best known codes is maintained in Reference [17], to which we compare the minimum weight of all codes constructed in this paper. An incidence structure is an ordered triple \(\mathcal{D}=(\mathcal{P},\mathcal{B},\mathcal{I})\) where \(\mathcal{P}\) and \(\mathcal{B}\) are non-empty disjoint sets and \(\mathcal{I}\subseteq\mathcal{P}\times\mathcal{B}\). The elements of the set \(\mathcal{P}\) are called points, the elements of the set \(\mathcal{B}\) are called blocks and \(\mathcal{I}\) is called an incidence relation. If \(|\mathcal{P}|=|\mathcal{B}|\) then the incidence structure is called symmetric. A \(t\)-\((v,k,\lambda)\) design is a finite incidence structure \(\mathcal{D}=(\mathcal{P},\mathcal{B},\mathcal{I})\) satisfying the following requirements:  
1. \(|\mathcal{P}|=v\),
2. every element of \(\mathcal{B}\) is incident with exactly \(k\) elements of \(\mathcal{P}\),
3. every \(t\) elements of \(\mathcal{P}\) are incident with exactly \(\lambda\) elements of \(\mathcal{B}\). A \(2\)-\((v,k,\lambda)\) design is called a block design. Note that this definition allows \(\mathcal{B}\) to be a multiset. If \(\mathcal{B}\) is a set then \(\mathcal{D}\) is called a simple design. If the design \(\mathcal{D}\) consists of \(k\) copies of some simple design \(\mathcal{D}^{\prime}\) then \(\mathcal{D}\) is non-simple design and it is denoted by \(\mathcal{D}=k\mathcal{D}^{\prime}\). We say that a \(t\)-\((v,k,\lambda)\) design \(\mathcal{D}\) is a quasi-symmetric design with intersection numbers \(x\) and \(y\ (x<y)\) if any two blocks of \(\mathcal{D}\) intersect in either \(x\) or \(y\) points. A graph is regular if all the vertices have the same degree. A connected graph \(\Gamma\) with diameter \(d\) is called distance-regular if there are integers \(b_{i},c_{i},i\geq 0\) such that, for any two vertices \(u,v\in\Gamma\) at distance \(d(u,v)=i\), there are exactly \(c_{i}\) neighbors of \(v\) at distance \(i-1\) from \(u\) and \(b_{i}\) neighbors of \(v\) at distance \(i+1\) from \(u\). A distance-regular graph is usually denoted by DRG. The graph \(\Gamma\) is a regular graph of valency \(k=b_{0}\). The numbers \(c_{i},b_{i},a_{i}\), where \(a_{i}=k-b_{i}-c_{i}\), \(i=0,\ldots,d\) is the number of neighbors of \(v\) at distance \(i\) from \(u\), are called the intersection numbers of \(\Gamma\).