Best Known (111−18, 111, s)-Nets in Base 4
(111−18, 111, 7285)-Net over F4 — Constructive and digital
Digital (93, 111, 7285)-net over F4, using
- 41 times duplication [i] based on digital (92, 110, 7285)-net over F4, using
- net defined by OOA [i] based on linear OOA(4110, 7285, F4, 18, 18) (dual of [(7285, 18), 131020, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4110, 65565, F4, 18) (dual of [65565, 65455, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- OA 9-folding and stacking [i] based on linear OA(4110, 65565, F4, 18) (dual of [65565, 65455, 19]-code), using
- net defined by OOA [i] based on linear OOA(4110, 7285, F4, 18, 18) (dual of [(7285, 18), 131020, 19]-NRT-code), using
(111−18, 111, 32783)-Net over F4 — Digital
Digital (93, 111, 32783)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4111, 32783, F4, 2, 18) (dual of [(32783, 2), 65455, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4111, 65566, F4, 18) (dual of [65566, 65455, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4111, 65567, F4, 18) (dual of [65567, 65456, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(45, 30, F4, 3) (dual of [30, 25, 4]-code or 30-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4111, 65567, F4, 18) (dual of [65567, 65456, 19]-code), using
- OOA 2-folding [i] based on linear OA(4111, 65566, F4, 18) (dual of [65566, 65455, 19]-code), using
(111−18, 111, large)-Net in Base 4 — Upper bound on s
There is no (93, 111, large)-net in base 4, because
- 16 times m-reduction [i] would yield (93, 95, large)-net in base 4, but