MinT - Best Known (51, 51+28, s)-Nets in Base 8

Best Known (51, 51+28, s)-Nets in Base 8

Digital (51, 79, 354)-net over F₈, using

9 times m-reduction [i] based on digital (51, 88, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 44, 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]

(51, 79, 516)-net in base 8, using

1 times m-reduction [i] based on (51, 80, 516)-net in base 8, using
- base change [i] based on digital (31, 60, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 30, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

Digital (51, 79, 699)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁹, 699, F₈, 28) (dual of [699, 620, 29]-code), using
- 619 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (5, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 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, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0, 1, 28 times 0, 1, 30 times 0, 1, 33 times 0, 1, 35 times 0, 1, 38 times 0, 1, 42 times 0, 1, 45 times 0, 1, 49 times 0) [i] based on linear OA(8²⁸, 29, F₈, 28) (dual of [29, 1, 29]-code or 29-arc in PG(27,8)), using
  - dual of repetition code with length 29 [i]

There is no (51, 79, 107730)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 220880 380875 193575 477766 955546 375575 724667 178598 451081 880549 448842 560904 > 8⁷⁹ [i]