Best Known (88, 88+17, s)-Nets in Base 3
(88, 88+17, 2463)-Net over F3 — Constructive and digital
Digital (88, 105, 2463)-net over F3, using
- 31 times duplication [i] based on digital (87, 104, 2463)-net over F3, using
- net defined by OOA [i] based on linear OOA(3104, 2463, F3, 17, 17) (dual of [(2463, 17), 41767, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3104, 19705, F3, 17) (dual of [19705, 19601, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(3104, 19705, F3, 17) (dual of [19705, 19601, 18]-code), using
- net defined by OOA [i] based on linear OOA(3104, 2463, F3, 17, 17) (dual of [(2463, 17), 41767, 18]-NRT-code), using
(88, 88+17, 9775)-Net over F3 — Digital
Digital (88, 105, 9775)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3105, 9775, F3, 2, 17) (dual of [(9775, 2), 19445, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3105, 9853, F3, 2, 17) (dual of [(9853, 2), 19601, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3105, 19706, F3, 17) (dual of [19706, 19601, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3104, 19705, F3, 17) (dual of [19705, 19601, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(382, 19683, F3, 14) (dual of [19683, 19601, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3104, 19705, F3, 17) (dual of [19705, 19601, 18]-code), using
- OOA 2-folding [i] based on linear OA(3105, 19706, F3, 17) (dual of [19706, 19601, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(3105, 9853, F3, 2, 17) (dual of [(9853, 2), 19601, 18]-NRT-code), using
(88, 88+17, 3000787)-Net in Base 3 — Upper bound on s
There is no (88, 105, 3000788)-net in base 3, because
- 1 times m-reduction [i] would yield (88, 104, 3000788)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 41 745593 598994 266003 870561 838685 898960 877381 601217 > 3104 [i]