Best Known (40, 40+17, s)-Nets in Base 7
(40, 40+17, 300)-Net over F7 — Constructive and digital
Digital (40, 57, 300)-net over F7, using
- net defined by OOA [i] based on linear OOA(757, 300, F7, 17, 17) (dual of [(300, 17), 5043, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(757, 2401, F7, 17) (dual of [2401, 2344, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(757, 2401, F7, 17) (dual of [2401, 2344, 18]-code), using
(40, 40+17, 1521)-Net over F7 — Digital
Digital (40, 57, 1521)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(757, 1521, F7, 17) (dual of [1521, 1464, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(757, 2401, F7, 17) (dual of [2401, 2344, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(757, 2401, F7, 17) (dual of [2401, 2344, 18]-code), using
(40, 40+17, 516679)-Net in Base 7 — Upper bound on s
There is no (40, 57, 516680)-net in base 7, because
- 1 times m-reduction [i] would yield (40, 56, 516680)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 211590 023997 081608 292358 457935 936484 012941 157377 > 756 [i]