MinT - Best Known (165−45, 165, s)-Nets in Base 8

Best Known (165−45, 165, s)-Nets in Base 8

(165−45, 165, 1026)-Net over F₈ — Constructive and digital

Digital (120, 165, 1026)-net over F₈, using

t-expansion [i] based on 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]

(165−45, 165, 6025)-Net over F₈ — Digital

Digital (120, 165, 6025)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−45, 165, 6980386)-Net in Base 8 — Upper bound on s

There is no (120, 165, 6980387)-net in base 8, because

1 times m-reduction [i] would yield (120, 164, 6980387)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12786 685527 980301 325867 104467 990687 195567 614803 933835 358149 291282 094923 817370 823176 504845 814746 210824 607724 284525 643098 009168 748600 154903 797188 854584 > 8¹⁶⁴ [i]