MinT - Best Known (162−59, 162, s)-Nets in Base 8

Best Known (162−59, 162, s)-Nets in Base 8

(162−59, 162, 368)-Net over F₈ — Constructive and digital

Digital (103, 162, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 30, 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 (73, 132, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 66, 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]

(162−59, 162, 576)-Net in Base 8 — Constructive

(103, 162, 576)-net in base 8, using

2 times m-reduction [i] based on (103, 164, 576)-net in base 8, using
- trace code for nets [i] based on (21, 82, 288)-net in base 64, using
  - 2 times m-reduction [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]

(162−59, 162, 1089)-Net over F₈ — Digital

Digital (103, 162, 1089)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−59, 162, 172061)-Net in Base 8 — Upper bound on s

There is no (103, 162, 172062)-net in base 8, because

1 times m-reduction [i] would yield (103, 161, 172062)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 24 974469 787555 413882 967384 345867 293048 194699 852687 042065 869670 275318 540857 898103 268317 555055 343498 528867 879439 138926 138139 258901 405581 881401 185199 > 8¹⁶¹ [i]