MinT - Best Known (70, 70+31, s)-Nets in Base 8

Best Known (70, 70+31, s)-Nets in Base 8

(70, 70+31, 400)-Net over F₈ — Constructive and digital

Digital (70, 101, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 25, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (45, 76, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 38, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(70, 70+31, 576)-Net in Base 8 — Constructive

(70, 101, 576)-net in base 8, using

5 times m-reduction [i] based on (70, 106, 576)-net in base 8, using
- trace code for nets [i] based on (17, 53, 288)-net in base 64, using
  - 3 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(70, 70+31, 1903)-Net over F₈ — Digital

Digital (70, 101, 1903)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+31, 962197)-Net in Base 8 — Upper bound on s

There is no (70, 101, 962198)-net in base 8, because

1 times m-reduction [i] would yield (70, 100, 962198)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 037040 373967 038406 420707 772272 590697 427337 520624 257912 945914 014538 409852 256720 367199 782856 > 8¹⁰⁰ [i]