MinT - Best Known (257−108, 257, s)-Nets in Base 4

Best Known (257−108, 257, s)-Nets in Base 4

(257−108, 257, 138)-Net over F₄ — Constructive and digital

Digital (149, 257, 138)-net over F₄, using

2 times m-reduction [i] based on digital (149, 259, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 76, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 183, 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]

(257−108, 257, 312)-Net over F₄ — Digital

Digital (149, 257, 312)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−108, 257, 5081)-Net in Base 4 — Upper bound on s

There is no (149, 257, 5082)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53813 208783 347887 969439 176376 084991 114619 026366 086821 167763 365707 194641 474730 504182 575076 547400 005143 759622 092906 051622 636908 201519 845009 967350 322331 075240 > 4²⁵⁷ [i]