MinT - Best Known (29, 48, s)-Nets in Base 8

Best Known (29, 48, s)-Nets in Base 8

Digital (29, 48, 256)-net over F₈, using

trace code for nets [i] based on digital (5, 24, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

Digital (29, 48, 287)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁴⁸, 287, F₈, 19) (dual of [287, 239, 20]-code), using
- 27 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 6 times 0, 1, 19 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]

(29, 48, 300)-net in base 8, using

There is no (29, 48, 30812)-net in base 8, because

1 times m-reduction [i] would yield (29, 47, 30812)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 787845 827028 041710 954888 808432 055786 019599 > 8⁴⁷ [i]