MinT - Best Known (156−79, 156, s)-Nets in Base 8

Best Known (156−79, 156, s)-Nets in Base 8

(156−79, 156, 130)-Net over F₈ — Constructive and digital

Digital (77, 156, 130)-net over F₈, using

t-expansion [i] based on digital (76, 156, 130)-net over F₈, using
- 16 times m-reduction [i] based on digital (76, 172, 130)-net over F₈, using
  - (u,Â u+v)-construction [i] based on
    1. digital (14, 62, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
    2. digital (14, 110, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈ (see above)

(156−79, 156, 239)-Net over F₈ — Digital

Digital (77, 156, 239)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(156−79, 156, 8516)-Net in Base 8 — Upper bound on s

There is no (77, 156, 8517)-net in base 8, because

1 times m-reduction [i] would yield (77, 155, 8517)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 95 316711 454545 192808 105115 814583 386703 108832 256476 661496 707913 600315 572756 536582 957906 781235 249603 386338 547474 998592 235106 209786 463369 270592 > 8¹⁵⁵ [i]