Best Known (219−39, 219, s)-Nets in Base 3
(219−39, 219, 896)-Net over F3 — Constructive and digital
Digital (180, 219, 896)-net over F3, using
- t-expansion [i] based on digital (178, 219, 896)-net over F3, using
- 1 times m-reduction [i] based on digital (178, 220, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 55, 224)-net over F81, using
- 1 times m-reduction [i] based on digital (178, 220, 896)-net over F3, using
(219−39, 219, 4707)-Net over F3 — Digital
Digital (180, 219, 4707)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3219, 4707, F3, 39) (dual of [4707, 4488, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3219, 6598, F3, 39) (dual of [6598, 6379, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- linear OA(3209, 6562, F3, 39) (dual of [6562, 6353, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to C([0,19]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3219, 6598, F3, 39) (dual of [6598, 6379, 40]-code), using
(219−39, 219, 1181727)-Net in Base 3 — Upper bound on s
There is no (180, 219, 1181728)-net in base 3, because
- 1 times m-reduction [i] would yield (180, 218, 1181728)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 102 905234 157202 015085 024745 813893 240760 309979 353850 710573 148334 950255 563674 311664 676587 768603 412404 757633 > 3218 [i]