MinT - Best Known (72, 72+33, s)-Nets in Base 8

Best Known (72, 72+33, s)-Nets in Base 8

(72, 72+33, 399)-Net over F₈ — Constructive and digital

Digital (72, 105, 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 (47, 80, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 40, 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]

(72, 72+33, 576)-Net in Base 8 — Constructive

(72, 105, 576)-net in base 8, using

5 times m-reduction [i] based on (72, 110, 576)-net in base 8, using
- trace code for nets [i] based on (17, 55, 288)-net in base 64, using
  - 1 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]

(72, 72+33, 1695)-Net over F₈ — Digital

Digital (72, 105, 1695)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+33, 720310)-Net in Base 8 — Upper bound on s

There is no (72, 105, 720311)-net in base 8, because

1 times m-reduction [i] would yield (72, 104, 720311)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8343 806016 663758 447219 557874 398705 757601 677770 751924 195999 088005 754535 350439 102474 183148 040003 > 8¹⁰⁴ [i]