MinT - Best Known (68, 68+65, s)-Nets in Base 4

Best Known (68, 68+65, s)-Nets in Base 4

(68, 68+65, 70)-Net over F₄ — Constructive and digital

Digital (68, 133, 70)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 35, 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, 98, 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]

(68, 68+65, 109)-Net over F₄ — Digital

Digital (68, 133, 109)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+65, 1271)-Net in Base 4 — Upper bound on s

There is no (68, 133, 1272)-net in base 4, because

1 times m-reduction [i] would yield (68, 132, 1272)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 29 657593 310347 355241 452044 738671 007728 056868 427433 075777 741776 330268 918847 262165 > 4¹³² [i]