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

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

(81, 81+81, 130)-Net over F₈ — Constructive and digital

Digital (81, 162, 130)-net over F₈, using

trace code for nets [i] based on digital (0, 81, 65)-net over F₆₄, using
- net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
    - the rational function field F₆₄(x) [i]
  - Niederreiter sequence [i]

(81, 81+81, 225)-Net in Base 8 — Constructive

(81, 162, 225)-net in base 8, using

2 times m-reduction [i] based on (81, 164, 225)-net in base 8, using
- base change [i] based on digital (40, 123, 225)-net over F₁₆, using
  - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
      - a function field by GarcÃa and Quoos [i]

(81, 81+81, 260)-Net over F₈ — Digital

Digital (81, 162, 260)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(81, 81+81, 9694)-Net in Base 8 — Upper bound on s

There is no (81, 162, 9695)-net in base 8, because

1 times m-reduction [i] would yield (81, 161, 9695)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 076936 249227 883260 623913 941284 550066 816929 308061 921476 456609 362690 055976 043735 274332 302872 355547 289260 025039 444249 714868 457381 521700 774012 399630 > 8¹⁶¹ [i]