MinT - Best Known (172−55, 172, s)-Nets in Base 8

Best Known (172−55, 172, s)-Nets in Base 8

(172−55, 172, 1026)-Net over F₈ — Constructive and digital

Digital (117, 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]

(172−55, 172, 2281)-Net over F₈ — Digital

Digital (117, 172, 2281)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−55, 172, 818219)-Net in Base 8 — Upper bound on s

There is no (117, 172, 818220)-net in base 8, because

1 times m-reduction [i] would yield (117, 171, 818220)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26816 266921 032097 065459 399068 870822 345009 232687 519925 885839 599659 018609 981420 800408 452294 656123 358896 229852 741288 216244 456620 617046 366374 452057 505895 395614 > 8¹⁷¹ [i]