MinT - Best Known (142−51, 142, s)-Nets in Base 8

Best Known (142−51, 142, s)-Nets in Base 8

(142−51, 142, 368)-Net over F₈ — Constructive and digital

Digital (91, 142, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 26, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [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]

(142−51, 142, 576)-Net in Base 8 — Constructive

(91, 142, 576)-net in base 8, using

8² times duplication [i] based on (89, 140, 576)-net in base 8, using
- trace code for nets [i] based on (19, 70, 288)-net in base 64, using
  - base change [i] based on digital (9, 60, 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]

(142−51, 142, 1047)-Net over F₈ — Digital

Digital (91, 142, 1047)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(142−51, 142, 180267)-Net in Base 8 — Upper bound on s

There is no (91, 142, 180268)-net in base 8, because

1 times m-reduction [i] would yield (91, 141, 180268)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 21 662527 077895 692292 366724 578556 795858 870218 228049 923637 948249 842552 064057 275658 547387 493866 518988 406737 316586 879861 825918 611378 > 8¹⁴¹ [i]