MinT - Best Known (245−90, 245, s)-Nets in Base 4

Best Known (245−90, 245, s)-Nets in Base 4

(245−90, 245, 160)-Net over F₄ — Constructive and digital

Digital (155, 245, 160)-net over F₄, using

8 times m-reduction [i] based on digital (155, 253, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 82, 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, 171, 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]

(245−90, 245, 196)-Net in Base 4 — Constructive

(155, 245, 196)-net in base 4, using

1 times m-reduction [i] based on (155, 246, 196)-net in base 4, using
- trace code for nets [i] based on (32, 123, 98)-net in base 16, using
  - 2 times m-reduction [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]

(245−90, 245, 466)-Net over F₄ — Digital

Digital (155, 245, 466)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−90, 245, 11104)-Net in Base 4 — Upper bound on s

There is no (155, 245, 11105)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3205 561709 309936 461192 090846 500296 669725 505314 992692 548142 568860 975823 980848 738836 290648 422708 112439 820873 649146 659641 861087 887083 365731 951134 032896 > 4²⁴⁵ [i]