MinT - Best Known (139, 139+84, s)-Nets in Base 4

Best Known (139, 139+84, s)-Nets in Base 4

(139, 139+84, 139)-Net over F₄ — Constructive and digital

Digital (139, 223, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 43, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (96, 180, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 90, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(139, 139+84, 394)-Net over F₄ — Digital

Digital (139, 223, 394)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139, 139+84, 8621)-Net in Base 4 — Upper bound on s

There is no (139, 223, 8622)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 181 836763 741152 885750 862497 777623 820019 050856 104147 760711 361837 555744 921631 108060 755715 811851 284408 150637 476178 990003 134884 338687 118510 > 4²²³ [i]