MinT - Best Known (142−44, 142, s)-Nets in Base 8

Best Known (142−44, 142, s)-Nets in Base 8

(142−44, 142, 419)-Net over F₈ — Constructive and digital

Digital (98, 142, 419)-net over F₈, using

3 times m-reduction [i] based on digital (98, 145, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 37, 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 (61, 108, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 54, 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]

(142−44, 142, 576)-Net in Base 8 — Constructive

(98, 142, 576)-net in base 8, using

t-expansion [i] based on (97, 142, 576)-net in base 8, using
- 12 times m-reduction [i] based on (97, 154, 576)-net in base 8, using
  - trace code for nets [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]

(142−44, 142, 2337)-Net over F₈ — Digital

Digital (98, 142, 2337)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(142−44, 142, 872536)-Net in Base 8 — Upper bound on s

There is no (98, 142, 872537)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 173 294360 309622 871836 295311 546260 622559 238827 020444 785238 997855 429489 510358 926458 648938 132055 051978 637022 060610 759130 504195 071204 > 8¹⁴² [i]