MinT - Best Known (178−138, 178, s)-Nets in Base 3

Best Known (178−138, 178, s)-Nets in Base 3

(178−138, 178, 42)-Net over F₃ — Constructive and digital

Digital (40, 178, 42)-net over F₃, using

t-expansion [i] based on digital (39, 178, 42)-net over F₃, using
- net from sequence [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(178−138, 178, 56)-Net over F₃ — Digital

Digital (40, 178, 56)-net over F₃, using

net from sequence [i] based on digital (40, 55)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
  - a curve from the manYPoints database [i]

(178−138, 178, 128)-Net in Base 3 — Upper bound on s

There is no (40, 178, 129)-net in base 3, because

62 times m-reduction [i] would yield (40, 116, 129)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹¹⁶, 129, S₃, 76), but
  - the linear programming bound shows that MÂ â‰¥ 10 043490 566221 756034 999262 067129 182567 059217 282309 049833 675189 / 424270 > 3¹¹⁶ [i]