MinT - Best Known (157, 250, s)-Nets in Base 4

Best Known (157, 250, s)-Nets in Base 4

(157, 250, 160)-Net over F₄ — Constructive and digital

Digital (157, 250, 160)-net over F₄, using

9 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]

(157, 250, 196)-Net in Base 4 — Constructive

(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]

(157, 250, 454)-Net over F₄ — Digital

Digital (157, 250, 454)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(157, 250, 10853)-Net in Base 4 — Upper bound on s

There is no (157, 250, 10854)-net in base 4, because

1 times m-reduction [i] would yield (157, 249, 10854)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 819523 293363 171641 152909 694264 425617 626206 672601 529071 645345 536354 497426 072030 787544 440372 485779 613860 591933 054518 556510 333026 739463 139671 176043 038368 > 4²⁴⁹ [i]