MinT - Best Known (56−40, 56, s)-Nets in Base 4

Best Known (56−40, 56, s)-Nets in Base 4

(56−40, 56, 33)-Net over F₄ — Constructive and digital

Digital (16, 56, 33)-net over F₄, using

t-expansion [i] based on digital (15, 56, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(56−40, 56, 36)-Net over F₄ — Digital

Digital (16, 56, 36)-net over F₄, using

net from sequence [i] based on digital (16, 35)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 16 and N(F) ≥ 36, using
  - Drinfeld modules of rank 1 [i]

(56−40, 56, 97)-Net in Base 4 — Upper bound on s

There is no (16, 56, 98)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁶, 98, S₄, 40), but
- the linear programming bound shows that MÂ â‰¥ 108 858270 389765 832134 899494 245555 240896 445146 215984 684712 894959 792949 029119 197184 / 19965 002432 218887 804054 871233 298540 608350 321175 > 4⁵⁶ [i]