MinT - Best Known (130−38, 130, s)-Nets in Base 8

Best Known (130−38, 130, s)-Nets in Base 8

(130−38, 130, 514)-Net over F₈ — Constructive and digital

Digital (92, 130, 514)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (21, 40, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 20, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (52, 90, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 45, 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]

(130−38, 130, 590)-Net in Base 8 — Constructive

(92, 130, 590)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 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. (72, 110, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 55, 288)-net in base 64, using
    - 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
      - base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
        net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(130−38, 130, 3136)-Net over F₈ — Digital

Digital (92, 130, 3136)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(130−38, 130, 1710610)-Net in Base 8 — Upper bound on s

There is no (92, 130, 1710611)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2521 730049 298895 828577 122232 768858 716063 616703 957445 776161 827670 339028 256273 375749 517520 738382 154037 818458 995523 615896 > 8¹³⁰ [i]