Best Known (188−27, 188, s)-Nets in Base 4
(188−27, 188, 20168)-Net over F4 — Constructive and digital
Digital (161, 188, 20168)-net over F4, using
- net defined by OOA [i] based on linear OOA(4188, 20168, F4, 27, 27) (dual of [(20168, 27), 544348, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4188, 262185, F4, 27) (dual of [262185, 261997, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4188, 262187, F4, 27) (dual of [262187, 261999, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(4181, 262144, F4, 27) (dual of [262144, 261963, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4188, 262187, F4, 27) (dual of [262187, 261999, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4188, 262185, F4, 27) (dual of [262185, 261997, 28]-code), using
(188−27, 188, 131093)-Net over F4 — Digital
Digital (161, 188, 131093)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4188, 131093, F4, 2, 27) (dual of [(131093, 2), 261998, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4188, 262186, F4, 27) (dual of [262186, 261998, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4188, 262187, F4, 27) (dual of [262187, 261999, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- linear OA(4181, 262144, F4, 27) (dual of [262144, 261963, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(26) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4188, 262187, F4, 27) (dual of [262187, 261999, 28]-code), using
- OOA 2-folding [i] based on linear OA(4188, 262186, F4, 27) (dual of [262186, 261998, 28]-code), using
(188−27, 188, large)-Net in Base 4 — Upper bound on s
There is no (161, 188, large)-net in base 4, because
- 25 times m-reduction [i] would yield (161, 163, large)-net in base 4, but