Best Known (191−30, 191, s)-Nets in Base 4
(191−30, 191, 4372)-Net over F4 — Constructive and digital
Digital (161, 191, 4372)-net over F4, using
- 42 times duplication [i] based on digital (159, 189, 4372)-net over F4, using
- net defined by OOA [i] based on linear OOA(4189, 4372, F4, 30, 30) (dual of [(4372, 30), 130971, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4189, 65580, F4, 30) (dual of [65580, 65391, 31]-code), using
- 5 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
- 5 times code embedding in larger space [i] based on linear OA(4184, 65575, F4, 30) (dual of [65575, 65391, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(4189, 65580, F4, 30) (dual of [65580, 65391, 31]-code), using
- net defined by OOA [i] based on linear OOA(4189, 4372, F4, 30, 30) (dual of [(4372, 30), 130971, 31]-NRT-code), using
(191−30, 191, 45823)-Net over F4 — Digital
Digital (161, 191, 45823)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4191, 45823, F4, 30) (dual of [45823, 45632, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 65590, F4, 30) (dual of [65590, 65399, 31]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4190, 65589, F4, 30) (dual of [65589, 65399, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [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(4137, 65536, F4, 23) (dual of [65536, 65399, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(413, 53, F4, 6) (dual of [53, 40, 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(29) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4190, 65589, F4, 30) (dual of [65589, 65399, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 65590, F4, 30) (dual of [65590, 65399, 31]-code), using
(191−30, 191, large)-Net in Base 4 — Upper bound on s
There is no (161, 191, large)-net in base 4, because
- 28 times m-reduction [i] would yield (161, 163, large)-net in base 4, but