MinT - Best Known (143−51, 143, s)-Nets in Base 4

Best Known (143−51, 143, s)-Nets in Base 4

(143−51, 143, 145)-Net over F₄ — Constructive and digital

Digital (92, 143, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 29, 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 (63, 114, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 57, 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]

(143−51, 143, 152)-Net in Base 4 — Constructive

(92, 143, 152)-net in base 4, using

1 times m-reduction [i] based on (92, 144, 152)-net in base 4, using
- trace code for nets [i] based on (20, 72, 76)-net in base 16, using
  - 3 times m-reduction [i] based on (20, 75, 76)-net in base 16, using
    - base change [i] based on digital (5, 60, 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]

(143−51, 143, 324)-Net over F₄ — Digital

Digital (92, 143, 324)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(143−51, 143, 8896)-Net in Base 4 — Upper bound on s

There is no (92, 143, 8897)-net in base 4, because

1 times m-reduction [i] would yield (92, 142, 8897)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 31 134291 410973 136474 425783 242951 868021 195493 360171 189528 778852 583568 019226 367769 850784 > 4¹⁴² [i]