MinT - Best Known (31, 78, s)-Nets in Base 128

Best Known (31, 78, s)-Nets in Base 128

(31, 78, 408)-Net over F₁₂₈ — Constructive and digital

Digital (31, 78, 408)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 26, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]
2. digital (5, 52, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(31, 78, 609)-Net over F₁₂₈ — Digital

Digital (31, 78, 609)-net over F₁₂₈, using

t-expansion [i] based on digital (30, 78, 609)-net over F₁₂₈, using
- net from sequence [i] based on digital (30, 608)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 30 and N(F) ≥ 609, using
    - a curve from the manYPoints database [i]

(31, 78, 841820)-Net in Base 128 — Upper bound on s

There is no (31, 78, 841821)-net in base 128, because

1 times m-reduction [i] would yield (31, 77, 841821)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 1 799609 173659 598406 949738 631032 519052 023266 972917 456149 749773 186912 504991 442911 145083 639887 216646 693203 697780 893514 502669 824842 360217 726114 403315 744564 143004 293112 > 128⁷⁷ [i]