MinT - Best Known (135−49, 135, s)-Nets in Base 8

Best Known (135−49, 135, s)-Nets in Base 8

(135−49, 135, 354)-Net over F₈ — Constructive and digital

Digital (86, 135, 354)-net over F₈, using

23 times m-reduction [i] based on digital (86, 158, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 79, 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]

(135−49, 135, 516)-Net in Base 8 — Constructive

(86, 135, 516)-net in base 8, using

1 times m-reduction [i] based on (86, 136, 516)-net in base 8, using
- base change [i] based on digital (52, 102, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 51, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(135−49, 135, 952)-Net over F₈ — Digital

Digital (86, 135, 952)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−49, 135, 154338)-Net in Base 8 — Upper bound on s

There is no (86, 135, 154339)-net in base 8, because

1 times m-reduction [i] would yield (86, 134, 154339)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 329320 663855 363563 878319 473918 522705 468904 623460 756231 645466 817479 935446 366567 491388 067880 563241 336425 405426 939846 228132 > 8¹³⁴ [i]