MinT - Best Known (70, 70+39, s)-Nets in Base 4

Best Known (70, 70+39, s)-Nets in Base 4

(70, 70+39, 135)-Net over F₄ — Constructive and digital

Digital (70, 109, 135)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 19, 5)-net over F₄, using
  - net from sequence [i] based on digital (0, 4)-sequence over F₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 0 and N(F) ≥ 5, using
      - the rational function field F₄(x) [i]
    - Niederreiter sequence [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]

(70, 70+39, 257)-Net over F₄ — Digital

Digital (70, 109, 257)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+39, 6972)-Net in Base 4 — Upper bound on s

There is no (70, 109, 6973)-net in base 4, because

1 times m-reduction [i] would yield (70, 108, 6973)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 105509 724373 200183 399903 917722 852071 499242 181203 167157 907029 913756 > 4¹⁰⁸ [i]