MinT - Best Known (81, 81+39, s)-Nets in Base 8

Best Known (81, 81+39, s)-Nets in Base 8

(81, 81+39, 399)-Net over F₈ — Constructive and digital

Digital (81, 120, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 28, 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 (53, 92, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 46, 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+39, 576)-Net in Base 8 — Constructive

(81, 120, 576)-net in base 8, using

6 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]

(81, 81+39, 1545)-Net over F₈ — Digital

Digital (81, 120, 1545)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+39, 513219)-Net in Base 8 — Upper bound on s

There is no (81, 120, 513220)-net in base 8, because

1 times m-reduction [i] would yield (81, 119, 513220)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 293578 571660 669191 096294 935858 014678 430781 015008 979171 773051 467094 463136 244812 133671 678491 869459 823761 963980 > 8¹¹⁹ [i]