MinT - Best Known (210−137, 210, s)-Nets in Base 4

Best Known (210−137, 210, s)-Nets in Base 4

(210−137, 210, 104)-Net over F₄ — Constructive and digital

Digital (73, 210, 104)-net over F₄, using

net from sequence [i] based on digital (73, 103)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
  - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(210−137, 210, 112)-Net over F₄ — Digital

Digital (73, 210, 112)-net over F₄, using

net from sequence [i] based on digital (73, 111)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
  - Drinfeld modules of rank 1 [i]

(210−137, 210, 563)-Net in Base 4 — Upper bound on s

There is no (73, 210, 564)-net in base 4, because

1 times m-reduction [i] would yield (73, 209, 564)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 710677 400839 421303 270956 257097 299797 522463 177818 643100 694814 724057 341755 691514 530547 583810 274887 992669 320871 516739 376083 728480 > 4²⁰⁹ [i]