MinT - Best Known (56, 56+117, s)-Nets in Base 8

Best Known (56, 56+117, s)-Nets in Base 8

(56, 56+117, 98)-Net over F₈ — Constructive and digital

Digital (56, 173, 98)-net over F₈, using

t-expansion [i] based on digital (37, 173, 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]

(56, 56+117, 144)-Net over F₈ — Digital

Digital (56, 173, 144)-net over F₈, using

t-expansion [i] based on digital (45, 173, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(56, 56+117, 1491)-Net in Base 8 — Upper bound on s

There is no (56, 173, 1492)-net in base 8, because

1 times m-reduction [i] would yield (56, 172, 1492)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 214903 357148 290732 345969 958679 755087 428382 229865 209621 462548 882467 452029 845394 575088 096981 961269 640700 901572 485854 846430 488768 065582 633514 344854 261482 415304 > 8¹⁷² [i]