Best Known (122, 122+27, s)-Nets in Base 3
(122, 122+27, 692)-Net over F3 — Constructive and digital
Digital (122, 149, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (109, 136, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 34, 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, 34, 172)-net over F81, using
- digital (0, 13, 4)-net over F3, using
(122, 122+27, 3374)-Net over F3 — Digital
Digital (122, 149, 3374)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3149, 3374, F3, 27) (dual of [3374, 3225, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3149, 6583, F3, 27) (dual of [6583, 6434, 28]-code), using
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 19, F3, 1) (dual of [19, 18, 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(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3149, 6583, F3, 27) (dual of [6583, 6434, 28]-code), using
(122, 122+27, 765968)-Net in Base 3 — Upper bound on s
There is no (122, 149, 765969)-net in base 3, because
- 1 times m-reduction [i] would yield (122, 148, 765969)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 41109 980930 772149 013286 963204 720119 171958 095565 129691 917860 986639 534835 > 3148 [i]