Best Known (216, 216+42, s)-Nets in Base 4
(216, 216+42, 3122)-Net over F4 — Constructive and digital
Digital (216, 258, 3122)-net over F4, using
- 44 times duplication [i] based on digital (212, 254, 3122)-net over F4, using
- net defined by OOA [i] based on linear OOA(4254, 3122, F4, 42, 42) (dual of [(3122, 42), 130870, 43]-NRT-code), using
- OA 21-folding and stacking [i] based on linear OA(4254, 65562, F4, 42) (dual of [65562, 65308, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4254, 65565, F4, 42) (dual of [65565, 65311, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(37) [i] based on
- linear OA(4249, 65536, F4, 42) (dual of [65536, 65287, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(4225, 65536, F4, 38) (dual of [65536, 65311, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(41) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(4254, 65565, F4, 42) (dual of [65565, 65311, 43]-code), using
- OA 21-folding and stacking [i] based on linear OA(4254, 65562, F4, 42) (dual of [65562, 65308, 43]-code), using
- net defined by OOA [i] based on linear OOA(4254, 3122, F4, 42, 42) (dual of [(3122, 42), 130870, 43]-NRT-code), using
(216, 216+42, 38774)-Net over F4 — Digital
Digital (216, 258, 38774)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4258, 38774, F4, 42) (dual of [38774, 38516, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4258, 65577, F4, 42) (dual of [65577, 65319, 43]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4256, 65575, F4, 42) (dual of [65575, 65319, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(36) [i] based on
- linear OA(4249, 65536, F4, 42) (dual of [65536, 65287, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(47, 39, F4, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(41) ⊂ Ce(36) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4256, 65575, F4, 42) (dual of [65575, 65319, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4258, 65577, F4, 42) (dual of [65577, 65319, 43]-code), using
(216, 216+42, large)-Net in Base 4 — Upper bound on s
There is no (216, 258, large)-net in base 4, because
- 40 times m-reduction [i] would yield (216, 218, large)-net in base 4, but