MinT - Best Known (128−85, 128, s)-Nets in Base 8

Best Known (128−85, 128, s)-Nets in Base 8

(128−85, 128, 98)-Net over F₈ — Constructive and digital

Digital (43, 128, 98)-net over F₈, using

t-expansion [i] based on digital (37, 128, 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]

(128−85, 128, 129)-Net over F₈ — Digital

Digital (43, 128, 129)-net over F₈, using

t-expansion [i] based on digital (38, 128, 129)-net over F₈, using
- net from sequence [i] based on digital (38, 128)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(128−85, 128, 1242)-Net in Base 8 — Upper bound on s

There is no (43, 128, 1243)-net in base 8, because

1 times m-reduction [i] would yield (43, 127, 1243)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 4 972918 294036 075359 779417 804540 381065 819760 869271 278982 232076 087339 518474 124928 596061 246925 130063 632438 855812 893410 > 8¹²⁷ [i]