MinT - Best Known (92−65, 92, s)-Nets in Base 4

Best Known (92−65, 92, s)-Nets in Base 4

(92−65, 92, 34)-Net over F₄ — Constructive and digital

Digital (27, 92, 34)-net over F₄, using

t-expansion [i] based on digital (21, 92, 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]

(92−65, 92, 42)-Net in Base 4 — Constructive

(27, 92, 42)-net in base 4, using

net from sequence [i] based on (27, 41)-sequence in base 4, using
- base expansion [i] based on digital (54, 41)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(92−65, 92, 55)-Net over F₄ — Digital

Digital (27, 92, 55)-net over F₄, using

t-expansion [i] based on digital (26, 92, 55)-net over F₄, using
- net from sequence [i] based on digital (26, 54)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 26 and N(F) ≥ 55, using
    - Drinfeld modules of rank 1 [i]

(92−65, 92, 133)-Net in Base 4 — Upper bound on s

There is no (27, 92, 134)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁹², 134, S₄, 65), but
- the linear programming bound shows that MÂ â‰¥ 258 409711 378905 818606 524061 223555 379325 323998 751428 743189 778694 319930 416238 972441 666542 108672 / 9 763312 671894 640560 347538 603720 618895 > 4⁹² [i]