Best Known (87, 87+17, s)-Nets in Base 3
(87, 87+17, 2463)-Net over F3 — Constructive and digital
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
(87, 87+17, 9036)-Net over F3 — Digital
Digital (87, 104, 9036)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3104, 9036, F3, 2, 17) (dual of [(9036, 2), 17968, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3104, 9852, F3, 2, 17) (dual of [(9852, 2), 19600, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3104, 19704, F3, 17) (dual of [19704, 19600, 18]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3104, 19705, F3, 17) (dual of [19705, 19601, 18]-code), using
- OOA 2-folding [i] based on linear OA(3104, 19704, F3, 17) (dual of [19704, 19600, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(3104, 9852, F3, 2, 17) (dual of [(9852, 2), 19600, 18]-NRT-code), using
(87, 87+17, 2615742)-Net in Base 3 — Upper bound on s
There is no (87, 104, 2615743)-net in base 3, because
- 1 times m-reduction [i] would yield (87, 103, 2615743)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13 915216 827431 423572 237744 702459 485015 680137 534449 > 3103 [i]