MinT - Best Known (70, 70+37, s)-Nets in Base 8

Best Known (70, 70+37, s)-Nets in Base 8

(70, 70+37, 368)-Net over F₈ — Constructive and digital

Digital (70, 107, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 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 (51, 88, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 44, 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]

(70, 70+37, 518)-Net in Base 8 — Constructive

(70, 107, 518)-net in base 8, using

1 times m-reduction [i] based on (70, 108, 518)-net in base 8, using
- base change [i] based on digital (43, 81, 518)-net over F₁₆, using
  - 1 times m-reduction [i] based on digital (43, 82, 518)-net over F₁₆, using
    - trace code for nets [i] based on digital (2, 41, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(70, 70+37, 1004)-Net over F₈ — Digital

Digital (70, 107, 1004)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+37, 224495)-Net in Base 8 — Upper bound on s

There is no (70, 107, 224496)-net in base 8, because

1 times m-reduction [i] would yield (70, 106, 224496)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 534028 137248 657417 968649 481417 662909 663966 213193 957409 712396 653269 863605 343147 083567 005880 564262 > 8¹⁰⁶ [i]