MinT - Best Known (165−55, 165, s)-Nets in Base 8

Best Known (165−55, 165, s)-Nets in Base 8

(165−55, 165, 513)-Net over F₈ — Constructive and digital

Digital (110, 165, 513)-net over F₈, using

base reduction for projective spaces (embedding PG(82,64) in PG(164,8)) for nets [i] based on digital (28, 83, 513)-net over F₆₄, using
- net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
    - the Hermitian function field over F₆₄ [i]

(165−55, 165, 576)-Net in Base 8 — Constructive

(110, 165, 576)-net in base 8, using

t-expansion [i] based on (108, 165, 576)-net in base 8, using
- 7 times m-reduction [i] based on (108, 172, 576)-net in base 8, using
  - trace code for nets [i] based on (22, 86, 288)-net in base 64, using
    - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
      - base change [i] based on digital (9, 78, 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]

(165−55, 165, 1748)-Net over F₈ — Digital

Digital (110, 165, 1748)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−55, 165, 477231)-Net in Base 8 — Upper bound on s

There is no (110, 165, 477232)-net in base 8, because

1 times m-reduction [i] would yield (110, 164, 477232)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12787 230856 061234 720861 814713 041422 615792 150120 626851 279789 169850 525368 482518 405840 470731 715963 444820 978073 973449 698049 270603 500217 547548 284113 637886 > 8¹⁶⁴ [i]