MinT - Best Known (82, 82+37, s)-Nets in Base 8

Best Known (82, 82+37, s)-Nets in Base 8

(82, 82+37, 402)-Net over F₈ — Constructive and digital

Digital (82, 119, 402)-net over F₈, using

1 times m-reduction [i] based on digital (82, 120, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 30, 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 (52, 90, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 45, 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]

(82, 82+37, 576)-Net in Base 8 — Constructive

(82, 119, 576)-net in base 8, using

t-expansion [i] based on (81, 119, 576)-net in base 8, using
- 7 times m-reduction [i] based on (81, 126, 576)-net in base 8, using
  - trace code for nets [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(82, 82+37, 1990)-Net over F₈ — Digital

Digital (82, 119, 1990)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+37, 898014)-Net in Base 8 — Upper bound on s

There is no (82, 119, 898015)-net in base 8, because

1 times m-reduction [i] would yield (82, 118, 898015)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 36696 505889 014999 712680 837665 690437 709018 838101 485155 689216 039936 590418 969985 393727 499990 354027 618987 187261 > 8¹¹⁸ [i]