MinT - Best Known (69−63, 69, s)-Nets in Base 4

Best Known (69−63, 69, s)-Nets in Base 4

(69−63, 69, 17)-Net over F₄ — Constructive and digital

Digital (6, 69, 17)-net over F₄, using

t-expansion [i] based on digital (5, 69, 17)-net over F₄, using
- net from sequence [i] based on digital (5, 16)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(69−63, 69, 20)-Net over F₄ — Digital

Digital (6, 69, 20)-net over F₄, using

net from sequence [i] based on digital (6, 19)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 6 and N(F) ≥ 20, using
  - Drinfeld modules of rank 1 [i]

(69−63, 69, 28)-Net in Base 4 — Upper bound on s

There is no (6, 69, 29)-net in base 4, because

14 times m-reduction [i] would yield (6, 55, 29)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4⁵⁵, 29, S₄, 2, 49), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 44134 523297 546034 842509 218798 370816 / 25 > 4⁵⁵ [i]