MinT - Best Known (154−52, 154, s)-Nets in Base 8

Best Known (154−52, 154, s)-Nets in Base 8

(154−52, 154, 400)-Net over F₈ — Constructive and digital

Digital (102, 154, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 36, 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 (66, 118, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 59, 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]

(154−52, 154, 576)-Net in Base 8 — Constructive

(102, 154, 576)-net in base 8, using

8 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]

(154−52, 154, 1538)-Net over F₈ — Digital

Digital (102, 154, 1538)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(154−52, 154, 336701)-Net in Base 8 — Upper bound on s

There is no (102, 154, 336702)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11 908631 425015 081801 250045 456646 674476 289410 884057 827645 477368 769735 463998 205040 789678 059854 215860 933590 091117 953192 163383 973655 191197 899010 > 8¹⁵⁴ [i]