Best Known (49−8, 49, s)-Nets in Base 7
(49−8, 49, 1441200)-Net over F7 — Constructive and digital
Digital (41, 49, 1441200)-net over F7, using
- net defined by OOA [i] based on linear OOA(749, 1441200, F7, 8, 8) (dual of [(1441200, 8), 11529551, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(749, 5764800, F7, 8) (dual of [5764800, 5764751, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(749, 5764800, F7, 8) (dual of [5764800, 5764751, 9]-code), using
(49−8, 49, 2882400)-Net over F7 — Digital
Digital (41, 49, 2882400)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(749, 2882400, F7, 2, 8) (dual of [(2882400, 2), 5764751, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(749, 5764800, F7, 8) (dual of [5764800, 5764751, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using
- OOA 2-folding [i] based on linear OA(749, 5764800, F7, 8) (dual of [5764800, 5764751, 9]-code), using
(49−8, 49, large)-Net in Base 7 — Upper bound on s
There is no (41, 49, large)-net in base 7, because
- 6 times m-reduction [i] would yield (41, 43, large)-net in base 7, but