MinT - Best Known (253−94, 253, s)-Nets in Base 4

Best Known (253−94, 253, s)-Nets in Base 4

(253−94, 253, 160)-Net over F₄ — Constructive and digital

Digital (159, 253, 160)-net over F₄, using

t-expansion [i] based on digital (157, 253, 160)-net over F₄, using
- 6 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]

(253−94, 253, 461)-Net over F₄ — Digital

Digital (159, 253, 461)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−94, 253, 10623)-Net in Base 4 — Upper bound on s

There is no (159, 253, 10624)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 209 711854 670717 300616 122135 427315 379197 212436 502270 446708 002739 799527 685295 353001 574612 941512 270112 632367 043674 182170 176595 476997 701041 278843 929002 529485 > 4²⁵³ [i]