Best Known (5, 5+7, s)-Nets in Base 7
(5, 5+7, 26)-Net over F7 — Constructive and digital
Digital (5, 12, 26)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 50)-net over F7, using
- net defined by OOA [i] based on linear OOA(74, 50, F7, 3, 3) (dual of [(50, 3), 146, 4]-NRT-code), using
- digital (1, 8, 13)-net over F7, using
- 5 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (1, 4, 50)-net over F7, using
(5, 5+7, 30)-Net over F7 — Digital
Digital (5, 12, 30)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(712, 30, F7, 7) (dual of [30, 18, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(712, 48, F7, 7) (dual of [48, 36, 8]-code), using
- the primitive narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(712, 48, F7, 7) (dual of [48, 36, 8]-code), using
(5, 5+7, 378)-Net in Base 7 — Upper bound on s
There is no (5, 12, 379)-net in base 7, because
- 1 times m-reduction [i] would yield (5, 11, 379)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 1983 162223 > 711 [i]