MinT - Best Known (248−90, 248, s)-Nets in Base 4

Best Known (248−90, 248, s)-Nets in Base 4

(248−90, 248, 160)-Net over F₄ — Constructive and digital

Digital (158, 248, 160)-net over F₄, using

t-expansion [i] based on digital (157, 248, 160)-net over F₄, using
- 11 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]

(248−90, 248, 208)-Net in Base 4 — Constructive

(158, 248, 208)-net in base 4, using

trace code for nets [i] based on (34, 124, 104)-net in base 16, using
- 1 times m-reduction [i] based on (34, 125, 104)-net in base 16, using
  - base change [i] based on digital (9, 100, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(248−90, 248, 492)-Net over F₄ — Digital

Digital (158, 248, 492)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−90, 248, 12183)-Net in Base 4 — Upper bound on s

There is no (158, 248, 12184)-net in base 4, because

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]