MinT - Best Known (57−37, 57, s)-Nets in Base 4

Best Known (57−37, 57, s)-Nets in Base 4

(57−37, 57, 33)-Net over F₄ — Constructive and digital

Digital (20, 57, 33)-net over F₄, using

t-expansion [i] based on digital (15, 57, 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]

(57−37, 57, 41)-Net over F₄ — Digital

Digital (20, 57, 41)-net over F₄, using

t-expansion [i] based on digital (18, 57, 41)-net over F₄, using
- net from sequence [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(57−37, 57, 173)-Net in Base 4 — Upper bound on s

There is no (20, 57, 174)-net in base 4, because

1 times m-reduction [i] would yield (20, 56, 174)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5249 631947 492790 676441 528977 865852 > 4⁵⁶ [i]