MinT - Best Known (71, 71+31, s)-Nets in Base 8

Best Known (71, 71+31, s)-Nets in Base 8

(71, 71+31, 402)-Net over F₈ — Constructive and digital

Digital (71, 102, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 26, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [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]

(71, 71+31, 576)-Net in Base 8 — Constructive

(71, 102, 576)-net in base 8, using

6 times m-reduction [i] based on (71, 108, 576)-net in base 8, using
- trace code for nets [i] based on (17, 54, 288)-net in base 64, using
  - 2 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]

(71, 71+31, 2039)-Net over F₈ — Digital

Digital (71, 102, 2039)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(71, 71+31, 1105276)-Net in Base 8 — Upper bound on s

There is no (71, 102, 1105277)-net in base 8, because

1 times m-reduction [i] would yield (71, 101, 1105277)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 16 296407 187144 157561 523647 300154 974778 175612 008657 068644 630907 082373 679182 547459 564692 028912 > 8¹⁰¹ [i]