MinT - Best Known (18, 18+55, s)-Nets in Base 4

Best Known (18, 18+55, s)-Nets in Base 4

(18, 18+55, 33)-Net over F₄ — Constructive and digital

Digital (18, 73, 33)-net over F₄, using

t-expansion [i] based on digital (15, 73, 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+55, 41)-Net over F₄ — Digital

Digital (18, 73, 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+55, 81)-Net in Base 4 — Upper bound on s

There is no (18, 73, 82)-net in base 4, because

1 times m-reduction [i] would yield (18, 72, 82)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁷², 82, S₄, 54), but
  - the linear programming bound shows that MÂ â‰¥ 258 868477 512236 179386 827675 994871 629856 557051 674624 / 8 133191 > 4⁷² [i]