MinT - Best Known (164−56, 164, s)-Nets in Base 8

Best Known (164−56, 164, s)-Nets in Base 8

(164−56, 164, 400)-Net over F₈ — Constructive and digital

Digital (108, 164, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 38, 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 (70, 126, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 63, 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]

(164−56, 164, 576)-Net in Base 8 — Constructive

(108, 164, 576)-net in base 8, using

8 times m-reduction [i] based on (108, 172, 576)-net in base 8, using
- trace code for nets [i] based on (22, 86, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
    - base change [i] based on digital (9, 78, 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]

(164−56, 164, 1530)-Net over F₈ — Digital

Digital (108, 164, 1530)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−56, 164, 314347)-Net in Base 8 — Upper bound on s

There is no (108, 164, 314348)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12787 359691 248091 263848 204604 674929 783629 297162 491367 689268 283615 597814 705401 976555 656797 724509 756383 577246 640465 716621 933946 423184 227336 438590 041064 > 8¹⁶⁴ [i]