MinT - Best Known (75, 75+39, s)-Nets in Base 4

Best Known (75, 75+39, s)-Nets in Base 4

(75, 75+39, 147)-Net over F₄ — Constructive and digital

Digital (75, 114, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 24, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (51, 90, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 45, 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]

(75, 75+39, 152)-Net in Base 4 — Constructive

(75, 114, 152)-net in base 4, using

2 times m-reduction [i] based on (75, 116, 152)-net in base 4, using
- trace code for nets [i] based on (17, 58, 76)-net in base 16, using
  - 2 times m-reduction [i] based on (17, 60, 76)-net in base 16, using
    - base change [i] based on digital (5, 48, 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]

(75, 75+39, 313)-Net over F₄ — Digital

Digital (75, 114, 313)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(75, 75+39, 10048)-Net in Base 4 — Upper bound on s

There is no (75, 114, 10049)-net in base 4, because

1 times m-reduction [i] would yield (75, 113, 10049)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 107 923754 265372 244287 812584 351269 660697 700180 233609 995145 842410 067904 > 4¹¹³ [i]