MinT - Best Known (162−56, 162, s)-Nets in Base 8

Best Known (162−56, 162, s)-Nets in Base 8

(162−56, 162, 389)-Net over F₈ — Constructive and digital

Digital (106, 162, 389)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 36, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using a function field by SÃ©mirat [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]

(162−56, 162, 576)-Net in Base 8 — Constructive

(106, 162, 576)-net in base 8, using

t-expansion [i] based on (105, 162, 576)-net in base 8, using
- 6 times m-reduction [i] based on (105, 168, 576)-net in base 8, using
  - trace code for nets [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(162−56, 162, 1421)-Net over F₈ — Digital

Digital (106, 162, 1421)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−56, 162, 270956)-Net in Base 8 — Upper bound on s

There is no (106, 162, 270957)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 199 802143 637114 689730 802395 926855 847142 595793 091509 585061 451561 866231 882561 730776 822204 945236 105770 596893 810748 374613 358164 377353 980990 257144 340974 > 8¹⁶² [i]