MinT - Best Known (53−37, 53, s)-Nets in Base 4

Best Known (53−37, 53, s)-Nets in Base 4

(53−37, 53, 33)-Net over F₄ — Constructive and digital

Digital (16, 53, 33)-net over F₄, using

t-expansion [i] based on digital (15, 53, 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]

(53−37, 53, 36)-Net over F₄ — Digital

Digital (16, 53, 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]

(53−37, 53, 114)-Net in Base 4 — Upper bound on s

There is no (16, 53, 115)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵³, 115, S₄, 37), but
- the linear programming bound shows that MÂ â‰¥ 2228 884102 741419 171563 165834 542871 015642 627518 277715 182772 794960 776775 241817 326574 230159 456820 575580 079378 861618 772055 855779 020800 / 26 684822 592939 485951 140315 267760 030177 231474 922984 007903 231959 034642 349539 832878 020392 293180 805179 > 4⁵³ [i]