MinT - Best Known (169−53, 169, s)-Nets in Base 8

Best Known (169−53, 169, s)-Nets in Base 8

(169−53, 169, 1026)-Net over F₈ — Constructive and digital

Digital (116, 169, 1026)-net over F₈, using

t-expansion [i] based on digital (114, 169, 1026)-net over F₈, using
- 3 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]

(169−53, 169, 2514)-Net over F₈ — Digital

Digital (116, 169, 2514)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−53, 169, 1031665)-Net in Base 8 — Upper bound on s

There is no (116, 169, 1031666)-net in base 8, because

1 times m-reduction [i] would yield (116, 168, 1031666)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 375061 464346 909224 413109 505020 890693 049067 743965 562960 412901 797291 992806 993780 380111 667812 024560 196575 020921 318718 728723 781652 321155 498577 108186 800240 > 8¹⁶⁸ [i]