MinT - Best Known (100, 100+51, s)-Nets in Base 8

Best Known (100, 100+51, s)-Nets in Base 8

(100, 100+51, 400)-Net over F₈ — Constructive and digital

Digital (100, 151, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 35, 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 (65, 116, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 58, 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]

(100, 100+51, 576)-Net in Base 8 — Constructive

(100, 151, 576)-net in base 8, using

7 times m-reduction [i] based on (100, 158, 576)-net in base 8, using
- trace code for nets [i] based on (21, 79, 288)-net in base 64, using
  - 5 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]

(100, 100+51, 1511)-Net over F₈ — Digital

Digital (100, 151, 1511)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(100, 100+51, 381109)-Net in Base 8 — Upper bound on s

There is no (100, 151, 381110)-net in base 8, because

1 times m-reduction [i] would yield (100, 150, 381110)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2907 418361 867446 611188 299253 812381 011495 477082 453843 765546 372779 183693 043347 618762 698887 189875 898912 392336 310425 680519 576231 393177 908476 > 8¹⁵⁰ [i]