MinT - Best Known (80−31, 80, s)-Nets in Base 8

Best Known (80−31, 80, s)-Nets in Base 8

(80−31, 80, 354)-Net over F₈ — Constructive and digital

Digital (49, 80, 354)-net over F₈, using

4 times m-reduction [i] based on digital (49, 84, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 42, 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]

(80−31, 80, 384)-Net in Base 8 — Constructive

(49, 80, 384)-net in base 8, using

trace code for nets [i] based on (9, 40, 192)-net in base 64, using
- 2 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
  - base change [i] based on digital (3, 36, 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]

(80−31, 80, 455)-Net over F₈ — Digital

Digital (49, 80, 455)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(80−31, 80, 52343)-Net in Base 8 — Upper bound on s

There is no (49, 80, 52344)-net in base 8, because

1 times m-reduction [i] would yield (49, 79, 52344)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 220857 575738 004099 304980 401381 612585 365130 337254 725985 794974 709084 935930 > 8⁷⁹ [i]