Best Known (168, 168+31, s)-Nets in Base 4
(168, 168+31, 4372)-Net over F4 — Constructive and digital
Digital (168, 199, 4372)-net over F4, using
- 44 times duplication [i] based on digital (164, 195, 4372)-net over F4, using
- net defined by OOA [i] based on linear OOA(4195, 4372, F4, 31, 31) (dual of [(4372, 31), 135337, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4195, 65581, F4, 31) (dual of [65581, 65386, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4195, 65586, F4, 31) (dual of [65586, 65391, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(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(30) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4195, 65586, F4, 31) (dual of [65586, 65391, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4195, 65581, F4, 31) (dual of [65581, 65386, 32]-code), using
- net defined by OOA [i] based on linear OOA(4195, 4372, F4, 31, 31) (dual of [(4372, 31), 135337, 32]-NRT-code), using
(168, 168+31, 50167)-Net over F4 — Digital
Digital (168, 199, 50167)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4199, 50167, F4, 31) (dual of [50167, 49968, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4199, 65590, F4, 31) (dual of [65590, 65391, 32]-code), using
- 4 times code embedding in larger space [i] based on linear OA(4195, 65586, F4, 31) (dual of [65586, 65391, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(4185, 65536, F4, 31) (dual of [65536, 65351, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(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(30) ⊂ Ce(24) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(4195, 65586, F4, 31) (dual of [65586, 65391, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4199, 65590, F4, 31) (dual of [65590, 65391, 32]-code), using
(168, 168+31, large)-Net in Base 4 — Upper bound on s
There is no (168, 199, large)-net in base 4, because
- 29 times m-reduction [i] would yield (168, 170, large)-net in base 4, but