MinT - Best Known (42, 42+72, s)-Nets in Base 3

Best Known (42, 42+72, s)-Nets in Base 3

(42, 42+72, 42)-Net over F₃ — Constructive and digital

Digital (42, 114, 42)-net over F₃, using

t-expansion [i] based on digital (39, 114, 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]

(42, 42+72, 56)-Net over F₃ — Digital

Digital (42, 114, 56)-net over F₃, using

t-expansion [i] based on digital (40, 114, 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]

(42, 42+72, 140)-Net in Base 3 — Upper bound on s

There is no (42, 114, 141)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹¹⁴, 141, S₃, 72), but
- the linear programming bound shows that MÂ â‰¥ 17712 898569 121247 955516 636418 953182 266284 226997 864009 328196 441696 418931 / 6573 090422 914675 > 3¹¹⁴ [i]