Best Known (116, 145, s)-Nets in Base 3
(116, 145, 688)-Net over F3 — Constructive and digital
Digital (116, 145, 688)-net over F3, using
- 31 times duplication [i] based on digital (115, 144, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 36, 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, 36, 172)-net over F81, using
(116, 145, 1889)-Net over F3 — Digital
Digital (116, 145, 1889)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3145, 1889, F3, 29) (dual of [1889, 1744, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3145, 2226, F3, 29) (dual of [2226, 2081, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(311, 39, F3, 5) (dual of [39, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(37, 14, F3, 5) (dual of [14, 7, 6]-code), using
- extended quadratic residue code Qe(14,3) [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3145, 2226, F3, 29) (dual of [2226, 2081, 30]-code), using
(116, 145, 244307)-Net in Base 3 — Upper bound on s
There is no (116, 145, 244308)-net in base 3, because
- 1 times m-reduction [i] would yield (116, 144, 244308)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 507 545736 735165 195184 156400 794198 399920 997756 348333 241885 098480 551609 > 3144 [i]