MinT - Best Known (18, 69, s)-Nets in Base 4

Best Known (18, 69, s)-Nets in Base 4

(18, 69, 33)-Net over F₄ — Constructive and digital

Digital (18, 69, 33)-net over F₄, using

t-expansion [i] based on digital (15, 69, 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, 69, 41)-Net over F₄ — Digital

Digital (18, 69, 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, 69, 83)-Net in Base 4 — Upper bound on s

There is no (18, 69, 84)-net in base 4, because

1 times m-reduction [i] would yield (18, 68, 84)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁶⁸, 84, S₄, 50), but
  - the linear programming bound shows that MÂ â‰¥ 30641 021233 608711 950166 966630 011186 880907 725233 979392 / 306664 536307 > 4⁶⁸ [i]