MinT - Best Known (109−34, 109, s)-Nets in Base 8

Best Known (109−34, 109, s)-Nets in Base 8

(109−34, 109, 400)-Net over F₈ — Constructive and digital

Digital (75, 109, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 27, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (48, 82, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 41, 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]

(109−34, 109, 576)-Net in Base 8 — Constructive

(75, 109, 576)-net in base 8, using

t-expansion [i] based on (73, 109, 576)-net in base 8, using
- 3 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
  - trace code for nets [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]

(109−34, 109, 1824)-Net over F₈ — Digital

Digital (75, 109, 1824)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−34, 109, 632762)-Net in Base 8 — Upper bound on s

There is no (75, 109, 632763)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 273 412359 010332 126832 917081 044621 270818 383865 914050 601530 061835 793759 576450 124235 812177 747199 513182 > 8¹⁰⁹ [i]