MinT - Best Known (51, 51+65, s)-Nets in Base 8

Best Known (51, 51+65, s)-Nets in Base 8

(51, 51+65, 98)-Net over F₈ — Constructive and digital

Digital (51, 116, 98)-net over F₈, using

t-expansion [i] based on digital (37, 116, 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]

(51, 51+65, 144)-Net over F₈ — Digital

Digital (51, 116, 144)-net over F₈, using

t-expansion [i] based on digital (45, 116, 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]

(51, 51+65, 3195)-Net in Base 8 — Upper bound on s

There is no (51, 116, 3196)-net in base 8, because

1 times m-reduction [i] would yield (51, 115, 3196)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 71 981655 209222 437212 776892 688164 755265 879626 230772 372429 535407 598707 720645 135548 506476 344919 154550 510893 > 8¹¹⁵ [i]