Best Known (149, 149+39, s)-Nets in Base 3
(149, 149+39, 688)-Net over F3 — Constructive and digital
Digital (149, 188, 688)-net over F3, using
- t-expansion [i] based on digital (148, 188, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 47, 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, 47, 172)-net over F81, using
(149, 149+39, 1855)-Net over F3 — Digital
Digital (149, 188, 1855)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3188, 1855, F3, 39) (dual of [1855, 1667, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(3188, 2209, F3, 39) (dual of [2209, 2021, 40]-code), using
- construction XX applied to Ce(39) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3169, 2187, F3, 37) (dual of [2187, 2018, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(31, 18, F3, 1) (dual of [18, 17, 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, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(39) ⊂ Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3188, 2209, F3, 39) (dual of [2209, 2021, 40]-code), using
(149, 149+39, 196798)-Net in Base 3 — Upper bound on s
There is no (149, 188, 196799)-net in base 3, because
- 1 times m-reduction [i] would yield (149, 187, 196799)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 166604 134859 669320 406584 814992 707954 648529 927096 756952 623457 189650 548660 544556 075710 314203 > 3187 [i]