Best Known (154, 154+41, s)-Nets in Base 3
(154, 154+41, 688)-Net over F3 — Constructive and digital
Digital (154, 195, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (154, 196, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 49, 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, 49, 172)-net over F81, using
(154, 154+41, 1783)-Net over F3 — Digital
Digital (154, 195, 1783)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3195, 1783, F3, 41) (dual of [1783, 1588, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3195, 2207, F3, 41) (dual of [2207, 2012, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [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(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(3195, 2207, F3, 41) (dual of [2207, 2012, 42]-code), using
(154, 154+41, 176321)-Net in Base 3 — Upper bound on s
There is no (154, 195, 176322)-net in base 3, because
- 1 times m-reduction [i] would yield (154, 194, 176322)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 364 389509 529925 991588 634565 062184 508896 633408 288442 941729 186956 160048 002542 619506 540175 279561 > 3194 [i]