MinT - Best Known (249−91, 249, s)-Nets in Base 4

Best Known (249−91, 249, s)-Nets in Base 4

(249−91, 249, 160)-Net over F₄ — Constructive and digital

Digital (158, 249, 160)-net over F₄, using

t-expansion [i] based on digital (157, 249, 160)-net over F₄, using
- 10 times m-reduction [i] based on digital (157, 259, 160)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (33, 84, 56)-net over F₄, using
      - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
    2. digital (73, 175, 104)-net over F₄, using
      - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(249−91, 249, 196)-Net in Base 4 — Constructive

(158, 249, 196)-net in base 4, using

t-expansion [i] based on (157, 249, 196)-net in base 4, using
- 1 times m-reduction [i] based on (157, 250, 196)-net in base 4, using
  - trace code for nets [i] based on (32, 125, 98)-net in base 16, using
    - base change [i] based on digital (7, 100, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(249−91, 249, 482)-Net over F₄ — Digital

Digital (158, 249, 482)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−91, 249, 12183)-Net in Base 4 — Upper bound on s

There is no (158, 249, 12184)-net in base 4, because

1 times m-reduction [i] would yield (158, 248, 12184)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 205295 574286 709284 638560 508795 454976 385460 722481 997758 899262 386327 329928 905345 458393 555386 093899 607578 934391 991858 259243 473376 640250 300839 963188 936760 > 4²⁴⁸ [i]