MinT - Best Known (54, 54+117, s)-Nets in Base 8

Best Known (54, 54+117, s)-Nets in Base 8

(54, 54+117, 98)-Net over F₈ — Constructive and digital

Digital (54, 171, 98)-net over F₈, using

t-expansion [i] based on digital (37, 171, 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]

(54, 54+117, 144)-Net over F₈ — Digital

Digital (54, 171, 144)-net over F₈, using

t-expansion [i] based on digital (45, 171, 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]

(54, 54+117, 1386)-Net in Base 8 — Upper bound on s

There is no (54, 171, 1387)-net in base 8, because

1 times m-reduction [i] would yield (54, 170, 1387)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3461 485240 206476 474270 807574 927607 675755 903769 196257 634912 801088 614125 050238 757625 803482 862390 842065 744001 528192 475700 474187 143927 322393 917202 980536 040628 > 8¹⁷⁰ [i]