MinT - Best Known (240−95, 240, s)-Nets in Base 4

Best Known (240−95, 240, s)-Nets in Base 4

(240−95, 240, 138)-Net over F₄ — Constructive and digital

Digital (145, 240, 138)-net over F₄, using

7 times m-reduction [i] based on digital (145, 247, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 72, 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, 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]

(240−95, 240, 139)-Net in Base 4 — Constructive

(145, 240, 139)-net in base 4, using

1 times m-reduction [i] based on (145, 241, 139)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (24, 72, 35)-net in base 4, using
    - net from sequence [i] based on (24, 34)-sequence in base 4, using
      - base expansion [i] based on digital (48, 34)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
  2. digital (73, 169, 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]

(240−95, 240, 356)-Net over F₄ — Digital

Digital (145, 240, 356)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−95, 240, 7016)-Net in Base 4 — Upper bound on s

There is no (145, 240, 7017)-net in base 4, because

1 times m-reduction [i] would yield (145, 239, 7017)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 781782 574554 824922 226475 496605 554398 772003 048896 488448 245139 243410 370331 248518 290077 531493 675556 722907 680389 864926 564816 502289 200088 324740 725440 > 4²³⁹ [i]