MinT - Best Known (18, 18+44, s)-Nets in Base 4

Best Known (18, 18+44, s)-Nets in Base 4

(18, 18+44, 33)-Net over F₄ — Constructive and digital

Digital (18, 62, 33)-net over F₄, using

t-expansion [i] based on digital (15, 62, 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+44, 41)-Net over F₄ — Digital

Digital (18, 62, 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+44, 106)-Net in Base 4 — Upper bound on s

There is no (18, 62, 107)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶², 107, S₄, 44), but
- the linear programming bound shows that MÂ â‰¥ 17622 602942 558177 133437 108590 576398 639747 070941 352929 752748 829629 227505 655642 080394 898835 505152 / 826 744207 227851 738915 816443 333621 359995 803052 054368 328125 > 4⁶² [i]