Best Known (67, 67+19, s)-Nets in Base 3
(67, 67+19, 464)-Net over F3 — Constructive and digital
Digital (67, 86, 464)-net over F3, using
- 32 times duplication [i] based on digital (65, 84, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 21, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 21, 116)-net over F81, using
(67, 67+19, 1073)-Net over F3 — Digital
Digital (67, 86, 1073)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(386, 1073, F3, 2, 19) (dual of [(1073, 2), 2060, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(386, 1097, F3, 2, 19) (dual of [(1097, 2), 2108, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(386, 2194, F3, 19) (dual of [2194, 2108, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(386, 2195, F3, 19) (dual of [2195, 2109, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(385, 2187, F3, 19) (dual of [2187, 2102, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(378, 2187, F3, 17) (dual of [2187, 2109, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(386, 2195, F3, 19) (dual of [2195, 2109, 20]-code), using
- OOA 2-folding [i] based on linear OA(386, 2194, F3, 19) (dual of [2194, 2108, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(386, 1097, F3, 2, 19) (dual of [(1097, 2), 2108, 20]-NRT-code), using
(67, 67+19, 66498)-Net in Base 3 — Upper bound on s
There is no (67, 86, 66499)-net in base 3, because
- 1 times m-reduction [i] would yield (67, 85, 66499)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 35920 511260 023982 475150 278908 605446 249783 > 385 [i]