MinT - Best Known (250−216, 250, s)-Nets in Base 4

Best Known (250−216, 250, s)-Nets in Base 4

(250−216, 250, 56)-Net over F₄ — Constructive and digital

Digital (34, 250, 56)-net over F₄, using

t-expansion [i] based on digital (33, 250, 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]

(250−216, 250, 65)-Net over F₄ — Digital

Digital (34, 250, 65)-net over F₄, using

t-expansion [i] based on digital (33, 250, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(250−216, 250, 120)-Net in Base 4 — Upper bound on s

There is no (34, 250, 121)-net in base 4, because

12 times m-reduction [i] would yield (34, 238, 121)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²³⁸, 121, S₄, 2, 204), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 40 582731 154107 899007 960763 987311 161974 731775 854644 624961 169040 475129 832604 656953 061563 979339 825395 625077 750104 578286 097640 823683 830747 708981 772288 / 205 > 4²³⁸ [i]