MinT - Best Known (75−29, 75, s)-Nets in Base 8

Best Known (75−29, 75, s)-Nets in Base 8

Digital (46, 75, 354)-net over F₈, using

3 times m-reduction [i] based on digital (46, 78, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 39, 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]

Digital (46, 75, 438)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁵, 438, F₈, 29) (dual of [438, 363, 30]-code), using
- 362 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 2, 2, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 26 times 0, 1, 29 times 0) [i] based on linear OA(8²⁹, 30, F₈, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,8)), using
  - dual of repetition code with length 30 [i]

There is no (46, 75, 51259)-net in base 8, because

1 times m-reduction [i] would yield (46, 74, 51259)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 741462 394323 631753 730645 708893 507582 988413 534705 291874 915310 188508 > 8⁷⁴ [i]