Best Known (178−30, 178, s)-Nets in Base 4
(178−30, 178, 4369)-Net over F4 — Constructive and digital
Digital (148, 178, 4369)-net over F4, using
- 41 times duplication [i] based on digital (147, 177, 4369)-net over F4, using
- net defined by OOA [i] based on linear OOA(4177, 4369, F4, 30, 30) (dual of [(4369, 30), 130893, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4177, 65535, F4, 30) (dual of [65535, 65358, 31]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(4177, 65535, F4, 30) (dual of [65535, 65358, 31]-code), using
- net defined by OOA [i] based on linear OOA(4177, 4369, F4, 30, 30) (dual of [(4369, 30), 130893, 31]-NRT-code), using
(178−30, 178, 30586)-Net over F4 — Digital
Digital (148, 178, 30586)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4178, 30586, F4, 2, 30) (dual of [(30586, 2), 60994, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4178, 32772, F4, 2, 30) (dual of [(32772, 2), 65366, 31]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4177, 32772, F4, 2, 30) (dual of [(32772, 2), 65367, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4177, 65544, F4, 30) (dual of [65544, 65367, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [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(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(40, 8, F4, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(4177, 65544, F4, 30) (dual of [65544, 65367, 31]-code), using
- 41 times duplication [i] based on linear OOA(4177, 32772, F4, 2, 30) (dual of [(32772, 2), 65367, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4178, 32772, F4, 2, 30) (dual of [(32772, 2), 65366, 31]-NRT-code), using
(178−30, 178, large)-Net in Base 4 — Upper bound on s
There is no (148, 178, large)-net in base 4, because
- 28 times m-reduction [i] would yield (148, 150, large)-net in base 4, but