Best Known (115−18, 115, s)-Nets in Base 3
(115−18, 115, 2190)-Net over F3 — Constructive and digital
Digital (97, 115, 2190)-net over F3, using
- net defined by OOA [i] based on linear OOA(3115, 2190, F3, 18, 18) (dual of [(2190, 18), 39305, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3115, 19710, F3, 18) (dual of [19710, 19595, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3115, 19716, F3, 18) (dual of [19716, 19601, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3115, 19716, F3, 18) (dual of [19716, 19601, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3115, 19710, F3, 18) (dual of [19710, 19595, 19]-code), using
(115−18, 115, 9858)-Net over F3 — Digital
Digital (97, 115, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3115, 9858, F3, 2, 18) (dual of [(9858, 2), 19601, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3115, 19716, F3, 18) (dual of [19716, 19601, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(3115, 19716, F3, 18) (dual of [19716, 19601, 19]-code), using
(115−18, 115, 2589818)-Net in Base 3 — Upper bound on s
There is no (97, 115, 2589819)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 7 395107 149489 001240 215553 362240 775117 705476 673100 655527 > 3115 [i]