**2. Cyclotomic numbers**

Cyclotomic numbers are one of the most vital objects in Number Theory. These numbers had been substantially utilized in Cryptography, Coding Theory and other branches of Information Theory. Thus, calculation of cyclotomic numbers, so called to as cyclotomic number problems, of various orders is one of the primary problems in Number Theory. Complete answers for cyclotomic number problem for *e* = 2 � 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 22, *l*, 2*l*, *l* 2 , 2*l* <sup>2</sup> with *l* an odd prime had been investigated by many authors see ([35–40] and the references there in). The section contains the generalized definition of cyclotomic numbers of order *e*, useful notations followed by properties of cyclotomic numbers of order 2*l* 2 . These properties play a vital role in determining which cyclotomic numbers of order 2*l* <sup>2</sup> are sufficient for the determination of all 4*l* <sup>4</sup> cyclotomic numbers of order 2*l* 2 . The section also examines the cyclotomic matrices of order 2*l* 2 .

## **2.1 Definition and notations**

Let *e*≥2 be an integer, and *p* � 1 mod ð Þ*e* an odd prime. One writes *p* ¼ *ek* þ 1 for some positive integer *k*. Let **F***<sup>p</sup>* be the finite field of *p* elements and let *γ* be a generator of the cyclic group **F**<sup>∗</sup> *<sup>p</sup>* . For 0≤ *a*, *b*≤*e* � 1, the cyclotomic number ð Þ *a*, *b <sup>e</sup>* of order *e* is defined as the number of solutions ð Þ *s*, *t* of the following:

$$
\gamma^{\varepsilon+a} + \gamma^{\varepsilon t+b} + \mathbb{1} \equiv \mathbb{0} \pmod{p}; \qquad \mathbf{0} \le \mathfrak{s}, t \le k - \mathbf{1}. \tag{1}
$$

#### **2.2 Properties of cyclotomic numbers of order 2***l* **2**

In this subsection, we recalled some elementary properties of cyclotomic numbers of order 2*l* <sup>2</sup> [38]. Let *<sup>p</sup>* � 1 mod2*<sup>l</sup>* <sup>2</sup> � � be a prime for an odd prime *l* and we write *p* ¼ 2*l* 2 *k* þ 1 for some positive integer *k*. It is clear that ð Þ *a*, *b* <sup>2</sup>*<sup>l</sup>* <sup>2</sup> ¼ *a*<sup>0</sup> , *b*<sup>0</sup> � � 2*l* 2 whenever *a* � *a*<sup>0</sup> mod2*l* <sup>2</sup> � � and *<sup>b</sup>* � *<sup>b</sup>*<sup>0</sup> mod2*<sup>l</sup>* <sup>2</sup> � � as well as ð Þ *<sup>a</sup>*, *<sup>b</sup>* <sup>2</sup>*<sup>l</sup>* <sup>2</sup> ¼ 2*l* <sup>2</sup> � *<sup>a</sup>*, *<sup>b</sup>* � *<sup>a</sup>* � � 2*l* <sup>2</sup> . These imply the following:

$$(a,b)\_{2^l} = \begin{cases} (b,a)\_{2^l} & \text{if } \ k \text{ is even,} \\\\ (b+l^2, a+l^2)\_{2^l} & \text{if } \ k \text{ is odd.} \end{cases} \tag{2}$$

Applying these facts, one can check that

$$\sum\_{a=0}^{2^{2}-1} \sum\_{b=0}^{2^{2}-1} (a,b)\_{2^{2}} = q - 2 \tag{3}$$

and

$$\sum\_{b=0}^{2l^2-1} (a,b)\_{2l^2} = k - n\_a,\tag{4}$$

where *na* is given by

$$n\_a = \begin{cases} 1 & \text{if } |a=0, \ 2 \mid k \text{ or } \text{if } |a=l^2, \ 2 \nmid k, \\ 0 & \text{otherwise.} \end{cases}$$

### **3. Cyclotomic matrices**

This section presents the procedure to determine cyclotomic matrices of order 2*l* <sup>2</sup> for prime *l*. We determine the equality relation of cyclotomic numbers and discuss how few of the cyclotomic numbers are enough for the construction of whole cyclotomic matrix. Further generators for a chosen value of *p* will be determined followed by the generation of a cyclotomic matrix. At every step, we have included a numerical example for the convenience to understand the procedure easily.

**Definition:-** Cyclotomic matrix of order 2*l* 2 , *l* be a prime, is a square matrix of order 2*l* 2 , whose entries are pair of solutions ð Þ *a*, *b* <sup>2</sup>*<sup>l</sup>* <sup>2</sup> ; 0 ≤*a*, *b*≤ 2*l* <sup>2</sup> � 1, of the Eq. (1).

For instance **Table 1** depicts a typical cyclotomic matrix of order 8 (assuming *l* ¼ 2). Whose construction steps have been given in the next subsection.
