MinT - Best Known (81, 81+54, s)-Nets in Base 8

Best Known (81, 81+54, s)-Nets in Base 8

(81, 81+54, 354)-Net over F₈ — Constructive and digital

Digital (81, 135, 354)-net over F₈, using

13 times m-reduction [i] based on digital (81, 148, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 74, 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]

(81, 81+54, 384)-Net in Base 8 — Constructive

(81, 135, 384)-net in base 8, using

1 times m-reduction [i] based on (81, 136, 384)-net in base 8, using
- trace code for nets [i] based on (13, 68, 192)-net in base 64, using
  - 2 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
    - base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(81, 81+54, 586)-Net over F₈ — Digital

Digital (81, 135, 586)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+54, 51122)-Net in Base 8 — Upper bound on s

There is no (81, 135, 51123)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 636265 106597 982203 560690 365274 588670 527337 652126 868026 295474 971102 075255 228431 401923 762966 922860 214336 713972 583253 762752 > 8¹³⁵ [i]