MinT - Best Known (144−57, 144, s)-Nets in Base 8

Best Known (144−57, 144, s)-Nets in Base 8

(144−57, 144, 354)-Net over F₈ — Constructive and digital

Digital (87, 144, 354)-net over F₈, using

16 times m-reduction [i] based on digital (87, 160, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 80, 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]

(144−57, 144, 384)-Net in Base 8 — Constructive

(87, 144, 384)-net in base 8, using

2 times m-reduction [i] based on (87, 146, 384)-net in base 8, using
- trace code for nets [i] based on (14, 73, 192)-net in base 64, using
  - 4 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
    - base change [i] based on digital (3, 66, 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]

(144−57, 144, 650)-Net over F₈ — Digital

Digital (87, 144, 650)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−57, 144, 66069)-Net in Base 8 — Upper bound on s

There is no (87, 144, 66070)-net in base 8, because

1 times m-reduction [i] would yield (87, 143, 66070)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1386 580508 838433 314197 103130 201264 732551 142637 836573 113589 456144 469225 716860 267461 909012 793811 601005 674919 963051 523966 982089 792192 > 8¹⁴³ [i]