MinT - Best Known (80−58, 80, s)-Nets in Base 4

Best Known (80−58, 80, s)-Nets in Base 4

(80−58, 80, 34)-Net over F₄ — Constructive and digital

Digital (22, 80, 34)-net over F₄, using

t-expansion [i] based on digital (21, 80, 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]

(80−58, 80, 44)-Net over F₄ — Digital

Digital (22, 80, 44)-net over F₄, using

t-expansion [i] based on digital (21, 80, 44)-net over F₄, using
- net from sequence [i] based on digital (21, 43)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 44, using
    - a curve from the manYPoints database [i]

(80−58, 80, 101)-Net in Base 4 — Upper bound on s

There is no (22, 80, 102)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁸⁰, 102, S₄, 58), but
- the linear programming bound shows that MÂ â‰¥ 581 746442 425291 342181 273363 255756 640248 201544 539357 172063 010816 / 311 561486 247011 > 4⁸⁰ [i]