MinT - Best Known (165−51, 165, s)-Nets in Base 8

Best Known (165−51, 165, s)-Nets in Base 8

(165−51, 165, 1026)-Net over F₈ — Constructive and digital

Digital (114, 165, 1026)-net over F₈, using

7 times m-reduction [i] based on digital (114, 172, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 86, 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]

(165−51, 165, 2684)-Net over F₈ — Digital

Digital (114, 165, 2684)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−51, 165, 1221216)-Net in Base 8 — Upper bound on s

There is no (114, 165, 1221217)-net in base 8, because

1 times m-reduction [i] would yield (114, 164, 1221217)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12786 885950 922036 844349 667876 624184 622517 081438 575493 620307 217657 814955 824213 664439 357591 350972 835546 272407 161921 287695 552487 531242 182472 557569 089024 > 8¹⁶⁴ [i]