Best Known (119, 119+27, s)-Nets in Base 3
(119, 119+27, 688)-Net over F3 — Constructive and digital
Digital (119, 146, 688)-net over F3, using
- t-expansion [i] based on digital (118, 146, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 37, 172)-net over F81, using
- 2 times m-reduction [i] based on digital (118, 148, 688)-net over F3, using
(119, 119+27, 3289)-Net over F3 — Digital
Digital (119, 146, 3289)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3146, 3289, F3, 2, 27) (dual of [(3289, 2), 6432, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3146, 6578, F3, 27) (dual of [6578, 6432, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3146, 6579, F3, 27) (dual of [6579, 6433, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3145, 6562, F3, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 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 C([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3146, 6579, F3, 27) (dual of [6579, 6433, 28]-code), using
- OOA 2-folding [i] based on linear OA(3146, 6578, F3, 27) (dual of [6578, 6432, 28]-code), using
(119, 119+27, 594434)-Net in Base 3 — Upper bound on s
There is no (119, 146, 594435)-net in base 3, because
- 1 times m-reduction [i] would yield (119, 145, 594435)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1522 597773 188273 569017 381709 446642 839199 913910 163112 805115 371021 160111 > 3145 [i]