MinT - Best Known (81−58, 81, s)-Nets in Base 4

Best Known (81−58, 81, s)-Nets in Base 4

(81−58, 81, 34)-Net over F₄ — Constructive and digital

Digital (23, 81, 34)-net over F₄, using

t-expansion [i] based on digital (21, 81, 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]

(81−58, 81, 45)-Net over F₄ — Digital

Digital (23, 81, 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]

(81−58, 81, 110)-Net in Base 4 — Upper bound on s

There is no (23, 81, 111)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁸¹, 111, S₄, 58), but
- the linear programming bound shows that MÂ â‰¥ 1 963481 420913 783424 763160 347413 099878 137248 244029 361670 539434 872648 237056 / 320334 367698 391203 714375 > 4⁸¹ [i]