MinT - Best Known (78, 78+69, s)-Nets in Base 8

Best Known (78, 78+69, s)-Nets in Base 8

(78, 78+69, 208)-Net over F₈ — Constructive and digital

Digital (78, 147, 208)-net over F₈, using

3 times m-reduction [i] based on digital (78, 150, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 75, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(78, 78+69, 225)-Net in Base 8 — Constructive

(78, 147, 225)-net in base 8, using

5 times m-reduction [i] based on (78, 152, 225)-net in base 8, using
- base change [i] based on digital (40, 114, 225)-net over F₁₆, using
  - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
      - a function field by GarcÃa and Quoos [i]

(78, 78+69, 312)-Net over F₈ — Digital

Digital (78, 147, 312)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+69, 14578)-Net in Base 8 — Upper bound on s

There is no (78, 147, 14579)-net in base 8, because

1 times m-reduction [i] would yield (78, 146, 14579)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 710423 600674 405549 850001 598411 426436 721006 980712 854200 650348 757261 891013 974235 686409 245874 405287 785883 196248 960870 877726 229107 336661 > 8¹⁴⁶ [i]