MinT - Best Known (69, 69+64, s)-Nets in Base 4

Best Known (69, 69+64, s)-Nets in Base 4

(69, 69+64, 71)-Net over F₄ — Constructive and digital

Digital (69, 133, 71)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 36, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (33, 97, 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]

(69, 69+64, 114)-Net over F₄ — Digital

Digital (69, 133, 114)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+64, 1329)-Net in Base 4 — Upper bound on s

There is no (69, 133, 1330)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 120 159570 697285 552039 167444 876360 445825 519854 374397 248373 503255 310650 715763 758007 > 4¹³³ [i]