MinT - Best Known (59−42, 59, s)-Nets in Base 4

Best Known (59−42, 59, s)-Nets in Base 4

(59−42, 59, 33)-Net over F₄ — Constructive and digital

Digital (17, 59, 33)-net over F₄, using

t-expansion [i] based on digital (15, 59, 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]

(59−42, 59, 40)-Net over F₄ — Digital

Digital (17, 59, 40)-net over F₄, using

net from sequence [i] based on digital (17, 39)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 17 and N(F) ≥ 40, using
  - Drinfeld modules of rank 1 [i]

(59−42, 59, 102)-Net in Base 4 — Upper bound on s

There is no (17, 59, 103)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁹, 103, S₄, 42), but
- the linear programming bound shows that MÂ â‰¥ 20 209278 630007 954817 311657 592716 406577 471235 077989 401051 636395 089582 131646 221319 183990 947510 747136 / 59 997456 042944 145499 206694 218235 767252 553003 337898 939171 700875 > 4⁵⁹ [i]