Best Known (116, 116+31, s)-Nets in Base 3
(116, 116+31, 640)-Net over F3 — Constructive and digital
Digital (116, 147, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (116, 148, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 37, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 37, 160)-net over F81, using
(116, 116+31, 1446)-Net over F3 — Digital
Digital (116, 147, 1446)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3147, 1446, F3, 31) (dual of [1446, 1299, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3147, 2208, F3, 31) (dual of [2208, 2061, 32]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3146, 2207, F3, 31) (dual of [2207, 2061, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3127, 2188, F3, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,15]) ⊂ C([0,13]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3146, 2207, F3, 31) (dual of [2207, 2061, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3147, 2208, F3, 31) (dual of [2208, 2061, 32]-code), using
(116, 116+31, 141472)-Net in Base 3 — Upper bound on s
There is no (116, 147, 141473)-net in base 3, because
- 1 times m-reduction [i] would yield (116, 146, 141473)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4567 950500 179650 704642 685366 923861 336868 500129 992734 291648 219032 616171 > 3146 [i]