Five families of the narrow-sense primitive BCH codes over finite fields

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. delta((a,b)) = aq(m)-1/q-1 + bq(m)-1/q2-1, where is even, 0 <= a <= q - 1, 1 <= b <= q - 1, 1 <= a + b <= q - 1 (2). (delta) over tilde ((a,b)) = aq(m-1) + (a + b)q(m-2) - 1, where is even, 0 <= a <= q-1, 1 <= b <= q-1, 1 <= a+b <= q-1. 3. delta((a,c)) = aq(m)-1/q-1 + cq(m-1)-1/q-1, where m >= 2, 0 <= a <= q-1, 1 <= c <= q-1, 1 <= a + c <= q-1. 4. delta((a,t))'=aq(m)-1/q-1 + q(m-1)-1/q-1-t, where m >= 3, 0 <= a <= q-2, a + 2 <= t <= q-1. 5. delta((a,c,t))''=aq(m)-1/q-1 + cq(m-1)-1/q-1-t, where m >= 3, 0 <= a <= q-3, 2 <= c <= q-, 1 <= a+c <= q-1, 1 <= t <= c-1. Moreover, we obtain the exact parameters of two subfamilies of BCH codes with designed distances (delta) over bar = bq(m)-1/q(2)-1 and delta(a,t)=(at+1)qm-1t(q-1) with even m, 1 <= a <= [q-2/t], 1 <= b <= q - 1, t>1 and t vertical bar(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.