MinT - Best Known (247−90, 247, s)-Nets in Base 4

Best Known (247−90, 247, s)-Nets in Base 4

(247−90, 247, 160)-Net over F₄ — Constructive and digital

Digital (157, 247, 160)-net over F₄, using

12 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]

(247−90, 247, 196)-Net in Base 4 — Constructive

(157, 247, 196)-net in base 4, using

3 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]

(247−90, 247, 483)-Net over F₄ — Digital

Digital (157, 247, 483)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−90, 247, 11812)-Net in Base 4 — Upper bound on s

There is no (157, 247, 11813)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51274 037374 407550 950713 360642 587180 585583 478745 622546 018163 533803 089614 935973 247938 990028 720057 288637 430297 379831 516348 449898 140896 852416 588957 441280 > 4²⁴⁷ [i]