MinT - Best Known (51, 51+31, s)-Nets in Base 8

Best Known (51, 51+31, s)-Nets in Base 8

(51, 51+31, 354)-Net over F₈ — Constructive and digital

Digital (51, 82, 354)-net over F₈, using

6 times m-reduction [i] based on digital (51, 88, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 44, 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]

(51, 51+31, 384)-Net in Base 8 — Constructive

(51, 82, 384)-net in base 8, using

2 times m-reduction [i] based on (51, 84, 384)-net in base 8, using
- trace code for nets [i] based on (9, 42, 192)-net in base 64, using
  - base change [i] based on digital (3, 36, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(51, 51+31, 521)-Net over F₈ — Digital

Digital (51, 82, 521)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+31, 69071)-Net in Base 8 — Upper bound on s

There is no (51, 82, 69072)-net in base 8, because

1 times m-reduction [i] would yield (51, 81, 69072)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 14 137208 390207 699369 377626 830180 769497 371339 150098 160131 127417 581504 284551 > 8⁸¹ [i]