MinT - Best Known (89, 89+57, s)-Nets in Base 8

Best Known (89, 89+57, s)-Nets in Base 8

(89, 89+57, 354)-Net over F₈ — Constructive and digital

Digital (89, 146, 354)-net over F₈, using

18 times m-reduction [i] based on digital (89, 164, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 82, 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]

(89, 89+57, 432)-Net in Base 8 — Constructive

(89, 146, 432)-net in base 8, using

trace code for nets [i] based on (16, 73, 216)-net in base 64, using
- 4 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
  - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(89, 89+57, 703)-Net over F₈ — Digital

Digital (89, 146, 703)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+57, 76652)-Net in Base 8 — Upper bound on s

There is no (89, 146, 76653)-net in base 8, because

1 times m-reduction [i] would yield (89, 145, 76653)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 88755 725101 586416 717616 088923 468806 631082 357587 205144 782107 205062 782909 534519 631274 396968 465680 570258 261264 114318 915788 737497 005630 > 8¹⁴⁵ [i]