MinT - Best Known (226−93, 226, s)-Nets in Base 4

Best Known (226−93, 226, s)-Nets in Base 4

(226−93, 226, 134)-Net over F₄ — Constructive and digital

Digital (133, 226, 134)-net over F₄, using

1 times m-reduction [i] based on digital (133, 227, 134)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 60, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 167, 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]

(226−93, 226, 297)-Net over F₄ — Digital

Digital (133, 226, 297)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−93, 226, 5246)-Net in Base 4 — Upper bound on s

There is no (133, 226, 5247)-net in base 4, because

1 times m-reduction [i] would yield (133, 225, 5247)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2924 536569 664156 941916 874632 205307 978877 020074 529402 641679 604552 062889 328193 982420 257776 179591 973803 690111 565429 305752 939605 610345 000307 > 4²²⁵ [i]