Best Known (66, 66+14, s)-Nets in Base 3
(66, 66+14, 942)-Net over F3 — Constructive and digital
Digital (66, 80, 942)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (59, 73, 938)-net over F3, using
- net defined by OOA [i] based on linear OOA(373, 938, F3, 14, 14) (dual of [(938, 14), 13059, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(373, 6566, F3, 14) (dual of [6566, 6493, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 6569, F3, 14) (dual of [6569, 6496, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(365, 6561, F3, 13) (dual of [6561, 6496, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 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(373, 6569, F3, 14) (dual of [6569, 6496, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(373, 6566, F3, 14) (dual of [6566, 6493, 15]-code), using
- net defined by OOA [i] based on linear OOA(373, 938, F3, 14, 14) (dual of [(938, 14), 13059, 15]-NRT-code), using
- digital (0, 7, 4)-net over F3, using
(66, 66+14, 3648)-Net over F3 — Digital
Digital (66, 80, 3648)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(380, 3648, F3, 14) (dual of [3648, 3568, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(380, 6569, F3, 14) (dual of [6569, 6489, 15]-code), using
- (u, u+v)-construction [i] based on
- linear OA(37, 8, F3, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,3)), using
- dual of repetition code with length 8 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(37, 8, F3, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(380, 6569, F3, 14) (dual of [6569, 6489, 15]-code), using
(66, 66+14, 479398)-Net in Base 3 — Upper bound on s
There is no (66, 80, 479399)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 147 808877 304461 461310 341494 045229 484307 > 380 [i]