MinT - Best Known (259−222, 259, s)-Nets in Base 4

Best Known (259−222, 259, s)-Nets in Base 4

(259−222, 259, 56)-Net over F₄ — Constructive and digital

Digital (37, 259, 56)-net over F₄, using

t-expansion [i] based on digital (33, 259, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(259−222, 259, 66)-Net over F₄ — Digital

Digital (37, 259, 66)-net over F₄, using

net from sequence [i] based on digital (37, 65)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
  - Drinfeld modules of rank 1 [i]

(259−222, 259, 130)-Net in Base 4 — Upper bound on s

There is no (37, 259, 131)-net in base 4, because

1 times m-reduction [i] would yield (37, 258, 131)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁵⁸, 131, S₄, 2, 221), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 8 580997 075163 262143 727375 998851 741521 586794 125179 131761 743079 323981 928979 247070 065153 199550 826818 193721 620389 239351 072546 402484 999645 804765 717535 363893 821440 / 37 > 4²⁵⁸ [i]