MinT - Best Known (67, 67+34, s)-Nets in Base 4

Best Known (67, 67+34, s)-Nets in Base 4

(67, 67+34, 145)-Net over F₄ — Constructive and digital

Digital (67, 101, 145)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 21, 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 (46, 80, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 40, 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]

(67, 67+34, 152)-Net in Base 4 — Constructive

(67, 101, 152)-net in base 4, using

4¹ times duplication [i] based on (66, 100, 152)-net in base 4, using
- t-expansion [i] based on (65, 100, 152)-net in base 4, using
  - trace code for nets [i] based on (15, 50, 76)-net in base 16, using
    - base change [i] based on digital (5, 40, 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]

(67, 67+34, 304)-Net over F₄ — Digital

Digital (67, 101, 304)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+34, 9018)-Net in Base 4 — Upper bound on s

There is no (67, 101, 9019)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 439586 597346 716800 379471 262154 638793 205829 733145 135364 216130 > 4¹⁰¹ [i]