It is an interesting problem to determine the parameters of BCH codes, due to their wide applications. In this paper, we determine the dimension and the Bose distance of five families of the narrow-sense primitive BCH codes with the following designed distances:

1. (1)

   \(\delta _{(a,b)}=a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1}\), where m is even, \(0\le a \le q-1\), \(1\le b \le q-1\), \(1\le a+b \le q-1\).

2. (2)

   \(\tilde{\delta }_{(a,b)}=aq^{m-1}+(a+b)q^{m-2}-1\), where m is even, \(0\le a \le q-1\), \(1\le b \le q-1\), \(1\le a+b \le q-1\).

3. (3)

   \({\delta _{(a,c)}}=a\frac{q^m-1}{q-1}+c\frac{q^{m-1}-1}{q-1}\), where \(m\ge 2\), \(0\le a \le q-1\), \(1\le c \le q-1\), \(1\le a+c \le q-1\).

4. (4)

   \({\delta }'_{(a,t)}=a\frac{q^{m}-1}{q-1}+\frac{q^{m-1}-1}{q-1}-t\), where \(m\ge 3\), \(0\le a \le q-2\), \(a+2\le t \le q-1\).

5. (5)

   \({\delta }''_{(a,c,t)}=a\frac{q^{m}-1}{q-1}+c\frac{q^{m-1}-1}{q-1}-t\), where \(m\ge 3\), \(0\le a \le q-3\), \(2\le c \le q-1\), \(1\le a+c \le q-1\), \(1\le t \le c-1\).

Moreover, we obtain the exact parameters of two subfamilies of BCH codes with designed distances \(\bar{\delta }= b\frac{q^m-1}{q^2-1}\) and \(\delta _{(a,t)}= (at+1)\frac{q^m-1}{t(q-1)}\) with even m, \(1\le a \le \big \lfloor \frac{q-2}{t}\big \rfloor \), \(1\le b\le q-1\), \(t>1\) and \(t|(q+1)\). Note that we get the narrow-sense primitive BCH codes with flexible designed distance as to a, b, c, t. Finally, we obtain a lot of the optimal or the best narrow-sense primitive BCH codes.

由于其广泛的应用，确定 BCH 码的参数是一个有趣的问题。在本文中，我们确定了具有以下设计距离的窄义原始BCH码的五个族的维数和Bose距离：

1. (1)

   \(\delta _{(a,b)}=a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1}\)，其中m是偶数，\(0\le a \le q-1\)，\(1\le b \le q-1\)，\(1\le a+b \le q-1\)。

2. (2)

   \(\tilde{\delta }_{(a,b)}=aq^{m-1}+(a+b)q^{m-2}-1\)，其中m是偶数，\(0 \le a \le q-1\)、\(1\le b \le q-1\)、\(1\le a+b \le q-1\)。

3. (3)

   \({\delta _{(a,c)}}=a\frac{q^m-1}{q-1}+c\frac{q^{m-1}-1}{q-1} \) , 其中\(m\ge 2\) , \(0\le a \le q-1\) , \(1\le c \le q-1\) , \(1\le a+c \乐 q-1\)。

4. (4)

   \({\delta }'_{(a,t)}=a\frac{q^{m}-1}{q-1}+\frac{q^{m-1}-1}{q- 1}-t\) , 其中\(m\ge 3\) , \(0\le a \le q-2\) , \(a+2\le t \le q-1\)。

5. (5)

   \({\delta }''_{(a,c,t)}=a\frac{q^{m}-1}{q-1}+c\frac{q^{m-1}-1 }{q-1}-t\) , 其中\(m\ge 3\) , \(0\le a \le q-3\) , \(2\le c \le q-1\) , \ (1\le a+c \le q-1\) , \(1\le t \le c-1\)。

此外，我们获得了设计距离\(\bar{\delta }= b\frac{q^m-1}{q^2-1}\)和\(\delta _ {(a,t)}= (at+1)\frac{q^m-1}{t(q-1)}\)与偶数m , \(1\le a \le \big \lfloor \frac {q-2}{t}\big \rfloor \)、\(1\le b\le q-1\)、\(t>1\)和\(t|(q+1)\)。请注意，我们得到了关于a、  b、  c、  t具有灵活设计距离的狭义原始 BCH 码。最后，我们得到了很多最优的或最好的狭义原始BCH码。