MinT - Best Known (78−33, 78, s)-Nets in Base 8

Best Known (78−33, 78, s)-Nets in Base 8

(78−33, 78, 256)-Net over F₈ — Constructive and digital

Digital (45, 78, 256)-net over F₈, using

2 times m-reduction [i] based on digital (45, 80, 256)-net over F₈, using
- trace code for nets [i] based on digital (5, 40, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(78−33, 78, 258)-Net in Base 8 — Constructive

(45, 78, 258)-net in base 8, using

trace code for nets [i] based on (6, 39, 129)-net in base 64, using
- 3 times m-reduction [i] based on (6, 42, 129)-net in base 64, using
  - base change [i] based on digital (0, 36, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(78−33, 78, 322)-Net over F₈ — Digital

Digital (45, 78, 322)-net over F₈, using

trace code for nets [i] based on digital (6, 39, 161)-net over F₆₄, using
- net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
    - a curve from the manYPoints database [i]

(78−33, 78, 21545)-Net in Base 8 — Upper bound on s

There is no (45, 78, 21546)-net in base 8, because

1 times m-reduction [i] would yield (45, 77, 21546)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3451 201501 816533 222931 983789 731740 696458 912609 174689 553812 538782 550898 > 8⁷⁷ [i]