MinT - Best Known (67, 67+63, s)-Nets in Base 4

Best Known (67, 67+63, s)-Nets in Base 4

(67, 67+63, 70)-Net over F₄ — Constructive and digital

Digital (67, 130, 70)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 34, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (33, 96, 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]

(67, 67+63, 110)-Net over F₄ — Digital

Digital (67, 130, 110)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+63, 1299)-Net in Base 4 — Upper bound on s

There is no (67, 130, 1300)-net in base 4, because

1 times m-reduction [i] would yield (67, 129, 1300)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 463386 928847 385602 691962 395361 685619 012874 933666 712616 503729 481862 221366 451392 > 4¹²⁹ [i]