MinT - Best Known (18, 18+45, s)-Nets in Base 4

Best Known (18, 18+45, s)-Nets in Base 4

(18, 18+45, 33)-Net over F₄ — Constructive and digital

Digital (18, 63, 33)-net over F₄, using

t-expansion [i] based on digital (15, 63, 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]

(18, 18+45, 41)-Net over F₄ — Digital

Digital (18, 63, 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]

(18, 18+45, 102)-Net in Base 4 — Upper bound on s

There is no (18, 63, 103)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶³, 103, S₄, 45), but
- the linear programming bound shows that MÂ â‰¥ 30 963907 201042 572946 047598 093595 047914 220745 773823 416057 277667 026891 964416 / 314660 526989 952486 919791 531363 400925 > 4⁶³ [i]