MinT - Best Known (51, 51+111, s)-Nets in Base 8

Best Known (51, 51+111, s)-Nets in Base 8

(51, 51+111, 98)-Net over F₈ — Constructive and digital

Digital (51, 162, 98)-net over F₈, using

t-expansion [i] based on digital (37, 162, 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+111, 144)-Net over F₈ — Digital

Digital (51, 162, 144)-net over F₈, using

t-expansion [i] based on digital (45, 162, 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+111, 1307)-Net in Base 8 — Upper bound on s

There is no (51, 162, 1308)-net in base 8, because

1 times m-reduction [i] would yield (51, 161, 1308)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 913278 583782 462776 756905 842817 740843 993003 692962 051169 284947 836088 226152 799972 157474 535952 164752 139283 339846 199936 390616 345620 320780 334089 609680 > 8¹⁶¹ [i]