MinT - Best Known (245−88, 245, s)-Nets in Base 4

Best Known (245−88, 245, s)-Nets in Base 4

(245−88, 245, 160)-Net over F₄ — Constructive and digital

Digital (157, 245, 160)-net over F₄, using

14 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]

(245−88, 245, 208)-Net in Base 4 — Constructive

(157, 245, 208)-net in base 4, using

1 times m-reduction [i] based on (157, 246, 208)-net in base 4, using
- trace code for nets [i] based on (34, 123, 104)-net in base 16, using
  - 2 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]

(245−88, 245, 505)-Net over F₄ — Digital

Digital (157, 245, 505)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−88, 245, 12911)-Net in Base 4 — Upper bound on s

There is no (157, 245, 12912)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3205 111898 704953 364906 846134 940336 372554 074216 357920 737750 404145 914336 551531 934043 606098 134002 285343 486320 066447 129966 619140 303697 916795 292122 274385 > 4²⁴⁵ [i]