MinT - Best Known (144−51, 144, s)-Nets in Base 8

Best Known (144−51, 144, s)-Nets in Base 8

(144−51, 144, 378)-Net over F₈ — Constructive and digital

Digital (93, 144, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 28, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [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]

(144−51, 144, 576)-Net in Base 8 — Constructive

(93, 144, 576)-net in base 8, using

2 times m-reduction [i] based on (93, 146, 576)-net in base 8, using
- trace code for nets [i] based on (20, 73, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(144−51, 144, 1135)-Net over F₈ — Digital

Digital (93, 144, 1135)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−51, 144, 212897)-Net in Base 8 — Upper bound on s

There is no (93, 144, 212898)-net in base 8, because

1 times m-reduction [i] would yield (93, 143, 212898)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1386 401201 336525 506336 510807 274446 523468 676312 701009 364934 770555 082453 026515 386987 980949 850094 930094 656096 679131 287130 648377 202388 > 8¹⁴³ [i]