MinT - Best Known (129−47, 129, s)-Nets in Base 8

Best Known (129−47, 129, s)-Nets in Base 8

(129−47, 129, 354)-Net over F₈ — Constructive and digital

Digital (82, 129, 354)-net over F₈, using

21 times m-reduction [i] based on digital (82, 150, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 75, 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]

(129−47, 129, 516)-Net in Base 8 — Constructive

(82, 129, 516)-net in base 8, using

8¹ times duplication [i] based on (81, 128, 516)-net in base 8, using
- base change [i] based on digital (49, 96, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 48, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(129−47, 129, 899)-Net over F₈ — Digital

Digital (82, 129, 899)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−47, 129, 142957)-Net in Base 8 — Upper bound on s

There is no (82, 129, 142958)-net in base 8, because

1 times m-reduction [i] would yield (82, 128, 142958)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 39 405173 944173 051115 000804 372484 500722 167679 222843 957396 809797 248067 657435 919269 480382 201684 203473 369491 244580 759672 > 8¹²⁸ [i]