Best Known (188, 188+35, s)-Nets in Base 4
(188, 188+35, 3858)-Net over F4 — Constructive and digital
Digital (188, 223, 3858)-net over F4, using
- 43 times duplication [i] based on digital (185, 220, 3858)-net over F4, using
- net defined by OOA [i] based on linear OOA(4220, 3858, F4, 35, 35) (dual of [(3858, 35), 134810, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4220, 65587, F4, 35) (dual of [65587, 65367, 36]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4219, 65586, F4, 35) (dual of [65586, 65367, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- 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(410, 50, F4, 5) (dual of [50, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4219, 65586, F4, 35) (dual of [65586, 65367, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(4220, 65587, F4, 35) (dual of [65587, 65367, 36]-code), using
- net defined by OOA [i] based on linear OOA(4220, 3858, F4, 35, 35) (dual of [(3858, 35), 134810, 36]-NRT-code), using
(188, 188+35, 49229)-Net over F4 — Digital
Digital (188, 223, 49229)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4223, 49229, F4, 35) (dual of [49229, 49006, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4223, 65590, F4, 35) (dual of [65590, 65367, 36]-code), using
- 4 times code embedding in larger space [i] based on linear OA(4219, 65586, F4, 35) (dual of [65586, 65367, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(4209, 65536, F4, 35) (dual of [65536, 65327, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- 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(410, 50, F4, 5) (dual of [50, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(4219, 65586, F4, 35) (dual of [65586, 65367, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4223, 65590, F4, 35) (dual of [65590, 65367, 36]-code), using
(188, 188+35, large)-Net in Base 4 — Upper bound on s
There is no (188, 223, large)-net in base 4, because
- 33 times m-reduction [i] would yield (188, 190, large)-net in base 4, but