MinT - Best Known (67−50, 67, s)-Nets in Base 4

Best Known (67−50, 67, s)-Nets in Base 4

(67−50, 67, 33)-Net over F₄ — Constructive and digital

Digital (17, 67, 33)-net over F₄, using

t-expansion [i] based on digital (15, 67, 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]

(67−50, 67, 40)-Net over F₄ — Digital

Digital (17, 67, 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]

(67−50, 67, 78)-Net in Base 4 — Upper bound on s

There is no (17, 67, 79)-net in base 4, because

1 times m-reduction [i] would yield (17, 66, 79)-net in base 4, but
- extracting embedded orthogonal array [i] would yield OA(4⁶⁶, 79, S₄, 49), but
  - the linear programming bound shows that MÂ â‰¥ 17 234654 248255 773502 866229 541038 208331 169774 174208 / 2465 450925 > 4⁶⁶ [i]