MinT - Best Known (139−50, 139, s)-Nets in Base 4

Best Known (139−50, 139, s)-Nets in Base 4

(139−50, 139, 140)-Net over F₄ — Constructive and digital

Digital (89, 139, 140)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 27, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (62, 112, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 56, 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]

(139−50, 139, 152)-Net in Base 4 — Constructive

(89, 139, 152)-net in base 4, using

1 times m-reduction [i] based on (89, 140, 152)-net in base 4, using
- trace code for nets [i] based on (19, 70, 76)-net in base 16, using
  - base change [i] based on digital (5, 56, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(139−50, 139, 307)-Net over F₄ — Digital

Digital (89, 139, 307)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−50, 139, 7529)-Net in Base 4 — Upper bound on s

There is no (89, 139, 7530)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 485857 062367 266436 158287 131934 332612 355371 773444 212960 559184 889063 361440 795094 592042 > 4¹³⁹ [i]