MinT - Best Known (47−19, 47, s)-Nets in Base 8

Best Known (47−19, 47, s)-Nets in Base 8

Digital (28, 47, 208)-net over F₈, using

3 times m-reduction [i] based on digital (28, 50, 208)-net over F₈, using
- trace code for nets [i] based on digital (3, 25, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(28, 47, 258)-net in base 8, using

Digital (28, 47, 266)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴⁷, 266, F₈, 19) (dual of [266, 219, 20]-code), using
- 7 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 6 times 0) [i] based on linear OA(8⁴⁶, 258, F₈, 19) (dual of [258, 212, 20]-code), using
  - trace code [i] based on linear OA(64²³, 129, F₆₄, 19) (dual of [129, 106, 20]-code), using
    - extended algebraic-geometric code AG_e(F,109P) [i] based on function field F/F₆₄ with g(F) = 4 and N(F) ≥ 129, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

There is no (28, 47, 24455)-net in base 8, because

1 times m-reduction [i] would yield (28, 46, 24455)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 348556 348768 692847 513889 199900 102019 720288 > 8⁴⁶ [i]