MinT - Best Known (117−41, 117, s)-Nets in Base 8

Best Known (117−41, 117, s)-Nets in Base 8

(117−41, 117, 368)-Net over F₈ — Constructive and digital

Digital (76, 117, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (55, 96, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 48, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(117−41, 117, 518)-Net in Base 8 — Constructive

(76, 117, 518)-net in base 8, using

1 times m-reduction [i] based on (76, 118, 518)-net in base 8, using
- trace code for nets [i] based on (17, 59, 259)-net in base 64, using
  - 1 times m-reduction [i] based on (17, 60, 259)-net in base 64, using
    - base change [i] based on digital (2, 45, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(117−41, 117, 1007)-Net over F₈ — Digital

Digital (76, 117, 1007)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(117−41, 117, 205165)-Net in Base 8 — Upper bound on s

There is no (76, 117, 205166)-net in base 8, because

1 times m-reduction [i] would yield (76, 116, 205166)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 573 412473 267122 325667 920365 035812 354434 054840 281659 300097 179050 546862 912933 411010 887356 529341 268104 729258 > 8¹¹⁶ [i]