MinT - Best Known (109−86, 109, s)-Nets in Base 4

Best Known (109−86, 109, s)-Nets in Base 4

(109−86, 109, 34)-Net over F₄ — Constructive and digital

Digital (23, 109, 34)-net over F₄, using

t-expansion [i] based on digital (21, 109, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(109−86, 109, 45)-Net over F₄ — Digital

Digital (23, 109, 45)-net over F₄, using

net from sequence [i] based on digital (23, 44)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 23 and N(F) ≥ 45, using
  - a curve from the manYPoints database [i]

(109−86, 109, 99)-Net in Base 4 — Upper bound on s

There is no (23, 109, 100)-net in base 4, because

19 times m-reduction [i] would yield (23, 90, 100)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁹⁰, 100, S₄, 67), but
  - the linear programming bound shows that MÂ â‰¥ 295023 781563 173995 900282 102890 760321 556810 705889 809622 106112 / 187663 > 4⁹⁰ [i]