MinT - Best Known (51, 51+119, s)-Nets in Base 8

Best Known (51, 51+119, s)-Nets in Base 8

(51, 51+119, 98)-Net over F₈ — Constructive and digital

Digital (51, 170, 98)-net over F₈, using

t-expansion [i] based on digital (37, 170, 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+119, 144)-Net over F₈ — Digital

Digital (51, 170, 144)-net over F₈, using

t-expansion [i] based on digital (45, 170, 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+119, 1221)-Net in Base 8 — Upper bound on s

There is no (51, 170, 1222)-net in base 8, because

1 times m-reduction [i] would yield (51, 169, 1222)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 419 012766 777169 743098 059806 143263 302270 356334 070333 950991 879258 513672 217493 416223 984390 687533 513696 202593 344473 591198 570839 469409 187735 832242 359605 680944 > 8¹⁶⁹ [i]