Best Known (76, 76+14, s)-Nets in Base 3
(76, 76+14, 2820)-Net over F3 — Constructive and digital
Digital (76, 90, 2820)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 8, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (68, 82, 2813)-net over F3, using
- net defined by OOA [i] based on linear OOA(382, 2813, F3, 14, 14) (dual of [(2813, 14), 39300, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(382, 19691, F3, 14) (dual of [19691, 19609, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(382, 19692, F3, 14) (dual of [19692, 19610, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- 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(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(382, 19692, F3, 14) (dual of [19692, 19610, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(382, 19691, F3, 14) (dual of [19691, 19609, 15]-code), using
- net defined by OOA [i] based on linear OOA(382, 2813, F3, 14, 14) (dual of [(2813, 14), 39300, 15]-NRT-code), using
- digital (1, 8, 7)-net over F3, using
(76, 76+14, 9859)-Net over F3 — Digital
Digital (76, 90, 9859)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(390, 9859, F3, 2, 14) (dual of [(9859, 2), 19628, 15]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(388, 9858, F3, 2, 14) (dual of [(9858, 2), 19628, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(388, 19716, F3, 14) (dual of [19716, 19628, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- 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(355, 19683, F3, 10) (dual of [19683, 19628, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(13) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(388, 19716, F3, 14) (dual of [19716, 19628, 15]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(388, 9858, F3, 2, 14) (dual of [(9858, 2), 19628, 15]-NRT-code), using
(76, 76+14, 2303051)-Net in Base 3 — Upper bound on s
There is no (76, 90, 2303052)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 8 727989 608987 743166 765897 193891 364248 140113 > 390 [i]