MinT - Best Known (153−50, 153, s)-Nets in Base 8

Best Known (153−50, 153, s)-Nets in Base 8

(153−50, 153, 419)-Net over F₈ — Constructive and digital

Digital (103, 153, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 39, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]
2. digital (64, 114, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 57, 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]

(153−50, 153, 576)-Net in Base 8 — Constructive

(103, 153, 576)-net in base 8, using

11 times m-reduction [i] based on (103, 164, 576)-net in base 8, using
- trace code for nets [i] based on (21, 82, 288)-net in base 64, using
  - 2 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]

(153−50, 153, 1828)-Net over F₈ — Digital

Digital (103, 153, 1828)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(153−50, 153, 489130)-Net in Base 8 — Upper bound on s

There is no (103, 153, 489131)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 488609 075851 134417 579366 638480 297198 800162 381309 008400 609198 839933 843101 625476 611823 098636 931995 906300 266003 958088 581574 324796 157868 129708 > 8¹⁵³ [i]