MinT - Best Known (160−58, 160, s)-Nets in Base 8

Best Known (160−58, 160, s)-Nets in Base 8

(160−58, 160, 368)-Net over F₈ — Constructive and digital

Digital (102, 160, 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 (72, 130, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 65, 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]

(160−58, 160, 576)-Net in Base 8 — Constructive

(102, 160, 576)-net in base 8, using

2 times m-reduction [i] based on (102, 162, 576)-net in base 8, using
- trace code for nets [i] based on (21, 81, 288)-net in base 64, using
  - 3 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]

(160−58, 160, 1105)-Net over F₈ — Digital

Digital (102, 160, 1105)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(160−58, 160, 160154)-Net in Base 8 — Upper bound on s

There is no (102, 160, 160155)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 121803 117574 871872 923666 082044 816000 212146 848399 951989 134068 118078 521162 610306 595254 183538 069922 465500 996004 001838 279741 040975 951756 620604 791360 > 8¹⁶⁰ [i]