MinT - Best Known (138−68, 138, s)-Nets in Base 4

Best Known (138−68, 138, s)-Nets in Base 4

(138−68, 138, 70)-Net over F₄ — Constructive and digital

Digital (70, 138, 70)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 37, 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, 101, 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]

(138−68, 138, 110)-Net over F₄ — Digital

Digital (70, 138, 110)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−68, 138, 1225)-Net in Base 4 — Upper bound on s

There is no (70, 138, 1226)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 122842 743743 873063 994264 275577 480898 193689 198864 356138 062759 452169 909910 005339 906584 > 4¹³⁸ [i]