Best Known (90−19, 90, s)-Nets in Base 3
(90−19, 90, 464)-Net over F3 — Constructive and digital
Digital (71, 90, 464)-net over F3, using
- 2 times m-reduction [i] based on digital (71, 92, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 23, 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, 23, 116)-net over F81, using
(90−19, 90, 1114)-Net over F3 — Digital
Digital (71, 90, 1114)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(390, 1114, F3, 19) (dual of [1114, 1024, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(390, 2207, F3, 19) (dual of [2207, 2117, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(385, 2188, F3, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(371, 2188, F3, 15) (dual of [2188, 2117, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(35, 19, F3, 3) (dual of [19, 14, 4]-code or 19-cap in PG(4,3)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(390, 2207, F3, 19) (dual of [2207, 2117, 20]-code), using
(90−19, 90, 108364)-Net in Base 3 — Upper bound on s
There is no (71, 90, 108365)-net in base 3, because
- 1 times m-reduction [i] would yield (71, 89, 108365)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 909476 227327 913928 996699 811916 026585 119867 > 389 [i]