MinT - Best Known (65, 65+103, s)-Nets in Base 8

Best Known (65, 65+103, s)-Nets in Base 8

(65, 65+103, 98)-Net over F₈ — Constructive and digital

Digital (65, 168, 98)-net over F₈, using

t-expansion [i] based on digital (37, 168, 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]

(65, 65+103, 144)-Net over F₈ — Digital

Digital (65, 168, 144)-net over F₈, using

t-expansion [i] based on digital (45, 168, 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]

(65, 65+103, 2537)-Net in Base 8 — Upper bound on s

There is no (65, 168, 2538)-net in base 8, because

1 times m-reduction [i] would yield (65, 167, 2538)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 566757 334447 072131 987434 099721 415182 186991 605662 356397 440740 321727 922640 065212 156566 127843 133032 303962 010251 791110 759881 704098 533589 105373 744183 973872 > 8¹⁶⁷ [i]