MinT - Best Known (254−98, 254, s)-Nets in Base 4

Best Known (254−98, 254, s)-Nets in Base 4

(254−98, 254, 160)-Net over F₄ — Constructive and digital

Digital (156, 254, 160)-net over F₄, using

2 times m-reduction [i] based on digital (156, 256, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 83, 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, 173, 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]

(254−98, 254, 407)-Net over F₄ — Digital

Digital (156, 254, 407)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−98, 254, 8375)-Net in Base 4 — Upper bound on s

There is no (156, 254, 8376)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 842 700952 218099 223220 321676 686882 174781 323558 358293 739053 392562 042093 347680 616642 282166 642759 034274 114501 893782 859423 334992 589039 331333 313937 909563 491449 > 4²⁵⁴ [i]