Best Known (46, 46+19, s)-Nets in Base 7
(46, 46+19, 267)-Net over F7 — Constructive and digital
Digital (46, 65, 267)-net over F7, using
- net defined by OOA [i] based on linear OOA(765, 267, F7, 19, 19) (dual of [(267, 19), 5008, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(765, 2404, F7, 19) (dual of [2404, 2339, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(765, 2405, F7, 19) (dual of [2405, 2340, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(765, 2401, F7, 19) (dual of [2401, 2336, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(761, 2401, F7, 18) (dual of [2401, 2340, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(70, 4, F7, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(765, 2405, F7, 19) (dual of [2405, 2340, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(765, 2404, F7, 19) (dual of [2404, 2339, 20]-code), using
(46, 46+19, 1807)-Net over F7 — Digital
Digital (46, 65, 1807)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(765, 1807, F7, 19) (dual of [1807, 1742, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(765, 2401, F7, 19) (dual of [2401, 2336, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(765, 2401, F7, 19) (dual of [2401, 2336, 20]-code), using
(46, 46+19, 706615)-Net in Base 7 — Upper bound on s
There is no (46, 65, 706616)-net in base 7, because
- 1 times m-reduction [i] would yield (46, 64, 706616)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 1 219775 333226 788614 067886 617212 072039 976736 481132 745425 > 764 [i]