MinT - Best Known (71, 71+32, s)-Nets in Base 8

Best Known (71, 71+32, s)-Nets in Base 8

(71, 71+32, 399)-Net over F₈ — Constructive and digital

Digital (71, 103, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 25, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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 (46, 78, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 39, 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+32, 576)-Net in Base 8 — Constructive

(71, 103, 576)-net in base 8, using

5 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+32, 1792)-Net over F₈ — Digital

Digital (71, 103, 1792)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(71, 71+32, 632522)-Net in Base 8 — Upper bound on s

There is no (71, 103, 632523)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1042 984132 878455 074513 164420 728662 897379 533968 729370 857239 233579 323071 739104 150443 635376 349212 > 8¹⁰³ [i]