MinT - Best Known (51, 72, s)-Nets in Base 8

Best Known (51, 72, s)-Nets in Base 8

(51, 72, 386)-Net over F₈ — Constructive and digital

Digital (51, 72, 386)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 20, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 10, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (31, 52, 256)-net over F₈, using
  - trace code for nets [i] based on digital (5, 26, 128)-net over F₆₄, using
    - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
        a function field by SÃ©mirat [i]

(51, 72, 576)-Net in Base 8 — Constructive

(51, 72, 576)-net in base 8, using

8² times duplication [i] based on (49, 70, 576)-net in base 8, using
- trace code for nets [i] based on (14, 35, 288)-net in base 64, using
  - base change [i] based on digital (9, 30, 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]

(51, 72, 2125)-Net over F₈ — Digital

Digital (51, 72, 2125)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 72, 1670381)-Net in Base 8 — Upper bound on s

There is no (51, 72, 1670382)-net in base 8, because

1 times m-reduction [i] would yield (51, 71, 1670382)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13164 088560 843804 897690 780216 213217 033834 075452 628342 706737 294486 > 8⁷¹ [i]