MinT - Best Known (172−57, 172, s)-Nets in Base 8

Best Known (172−57, 172, s)-Nets in Base 8

(172−57, 172, 1026)-Net over F₈ — Constructive and digital

Digital (115, 172, 1026)-net over F₈, using

t-expansion [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]

(172−57, 172, 1870)-Net over F₈ — Digital

Digital (115, 172, 1870)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−57, 172, 528679)-Net in Base 8 — Upper bound on s

There is no (115, 172, 528680)-net in base 8, because

1 times m-reduction [i] would yield (115, 171, 528680)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26816 832817 272256 547916 053956 601722 861274 267917 381408 905568 136004 102137 028874 757500 431891 629499 212961 341627 706698 451553 199388 925584 746240 153283 563468 115852 > 8¹⁷¹ [i]