MinT - Best Known (254−97, 254, s)-Nets in Base 4

Best Known (254−97, 254, s)-Nets in Base 4

(254−97, 254, 160)-Net over F₄ — Constructive and digital

Digital (157, 254, 160)-net over F₄, using

5 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]

(254−97, 254, 421)-Net over F₄ — Digital

Digital (157, 254, 421)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−97, 254, 9272)-Net in Base 4 — Upper bound on s

There is no (157, 254, 9273)-net in base 4, because

1 times m-reduction [i] would yield (157, 253, 9273)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 210 353579 888634 452246 258985 668800 330567 945078 026222 540776 675500 769164 915968 817912 848563 625850 619501 456143 894148 950142 216048 757857 223782 089370 883233 540635 > 4²⁵³ [i]