MinT - Best Known (254−106, 254, s)-Nets in Base 4

Best Known (254−106, 254, s)-Nets in Base 4

(254−106, 254, 138)-Net over F₄ — Constructive and digital

Digital (148, 254, 138)-net over F₄, using

2 times m-reduction [i] based on digital (148, 256, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 75, 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, 181, 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]

(254−106, 254, 315)-Net over F₄ — Digital

Digital (148, 254, 315)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−106, 254, 5228)-Net in Base 4 — Upper bound on s

There is no (148, 254, 5229)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 838 509050 668338 092016 575316 976741 068661 559583 186157 626245 029424 544223 657556 708680 186397 742575 274913 293794 008310 997724 001041 708091 293466 886914 997824 598352 > 4²⁵⁴ [i]