MinT - Best Known (42, 42+81, s)-Nets in Base 8

Best Known (42, 42+81, s)-Nets in Base 8

(42, 42+81, 98)-Net over F₈ — Constructive and digital

Digital (42, 123, 98)-net over F₈, using

t-expansion [i] based on digital (37, 123, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(42, 42+81, 129)-Net over F₈ — Digital

Digital (42, 123, 129)-net over F₈, using

t-expansion [i] based on digital (38, 123, 129)-net over F₈, using
- net from sequence [i] based on digital (38, 128)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(42, 42+81, 1254)-Net in Base 8 — Upper bound on s

There is no (42, 123, 1255)-net in base 8, because

1 times m-reduction [i] would yield (42, 122, 1255)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 151 895126 415807 750071 270336 510675 515535 637940 876517 810372 338518 274501 470152 440815 995589 867403 598041 252775 941416 > 8¹²² [i]