Best Known (186−28, 186, s)-Nets in Base 4
(186−28, 186, 4696)-Net over F4 — Constructive and digital
Digital (158, 186, 4696)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (4, 18, 15)-net over F4, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 4 and N(F) ≥ 15, using
- net from sequence [i] based on digital (4, 14)-sequence over F4, using
- digital (140, 168, 4681)-net over F4, using
- net defined by OOA [i] based on linear OOA(4168, 4681, F4, 28, 28) (dual of [(4681, 28), 130900, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(4168, 65534, F4, 28) (dual of [65534, 65366, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4168, 65535, F4, 28) (dual of [65535, 65367, 29]-code), using
- 1 times truncation [i] based on linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 1 times truncation [i] based on linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4168, 65535, F4, 28) (dual of [65535, 65367, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(4168, 65534, F4, 28) (dual of [65534, 65366, 29]-code), using
- net defined by OOA [i] based on linear OOA(4168, 4681, F4, 28, 28) (dual of [(4681, 28), 130900, 29]-NRT-code), using
- digital (4, 18, 15)-net over F4, using
(186−28, 186, 65604)-Net over F4 — Digital
Digital (158, 186, 65604)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4186, 65604, F4, 28) (dual of [65604, 65418, 29]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4182, 65597, F4, 28) (dual of [65597, 65415, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(413, 61, F4, 6) (dual of [61, 48, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4182, 65600, F4, 26) (dual of [65600, 65418, 27]-code), using Gilbert–Varšamov bound and bm = 4182 > Vbs−1(k−1) = 143841 756290 016350 169662 786892 659721 213183 918115 845900 235006 828617 848740 709183 661310 225110 542099 256145 980430 [i]
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(4182, 65597, F4, 28) (dual of [65597, 65415, 29]-code), using
- construction X with Varšamov bound [i] based on
(186−28, 186, large)-Net in Base 4 — Upper bound on s
There is no (158, 186, large)-net in base 4, because
- 26 times m-reduction [i] would yield (158, 160, large)-net in base 4, but