MinT - Best Known (42−27, 42, s)-Nets in Base 81

Best Known (42−27, 42, s)-Nets in Base 81

(42−27, 42, 224)-Net over F₈₁ — Constructive and digital

Digital (15, 42, 224)-net over F₈₁, using

t-expansion [i] based on digital (13, 42, 224)-net over F₈₁, using
- net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
    - a function field by SÃ©mirat [i]

(42−27, 42, 298)-Net over F₈₁ — Digital

Digital (15, 42, 298)-net over F₈₁, using

t-expansion [i] based on digital (12, 42, 298)-net over F₈₁, using
- net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
    - a curve from the manYPoints database [i]

(42−27, 42, 74019)-Net in Base 81 — Upper bound on s

There is no (15, 42, 74020)-net in base 81, because

1 times m-reduction [i] would yield (15, 41, 74020)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 1 769691 550525 139625 128865 138458 521086 137412 096510 244842 084368 419314 396707 892801 > 81⁴¹ [i]