Best Known (186−30, 186, s)-Nets in Base 4
(186−30, 186, 4371)-Net over F4 — Constructive and digital
Digital (156, 186, 4371)-net over F4, using
- 44 times duplication [i] based on digital (152, 182, 4371)-net over F4, using
- net defined by OOA [i] based on linear OOA(4182, 4371, F4, 30, 30) (dual of [(4371, 30), 130948, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4182, 65565, F4, 30) (dual of [65565, 65383, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(25) [i] based on
- linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(29) ⊂ Ce(25) [i] based on
- OA 15-folding and stacking [i] based on linear OA(4182, 65565, F4, 30) (dual of [65565, 65383, 31]-code), using
- net defined by OOA [i] based on linear OOA(4182, 4371, F4, 30, 30) (dual of [(4371, 30), 130948, 31]-NRT-code), using
(186−30, 186, 35769)-Net over F4 — Digital
Digital (156, 186, 35769)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4186, 35769, F4, 30) (dual of [35769, 35583, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4186, 65577, F4, 30) (dual of [65577, 65391, 31]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4184, 65575, F4, 30) (dual of [65575, 65391, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(24) [i] based on
- linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(29) ⊂ Ce(24) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4184, 65575, F4, 30) (dual of [65575, 65391, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4186, 65577, F4, 30) (dual of [65577, 65391, 31]-code), using
(186−30, 186, large)-Net in Base 4 — Upper bound on s
There is no (156, 186, large)-net in base 4, because
- 28 times m-reduction [i] would yield (156, 158, large)-net in base 4, but