MinT - Best Known (260−108, 260, s)-Nets in Base 4

Best Known (260−108, 260, s)-Nets in Base 4

(260−108, 260, 138)-Net over F₄ — Constructive and digital

Digital (152, 260, 138)-net over F₄, using

t-expansion [i] based on digital (149, 260, 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, 184, 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]

(260−108, 260, 139)-Net in Base 4 — Constructive

(152, 260, 139)-net in base 4, using

t-expansion [i] based on (151, 260, 139)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (24, 78, 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, 182, 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]

(260−108, 260, 327)-Net over F₄ — Digital

Digital (152, 260, 327)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(260−108, 260, 5492)-Net in Base 4 — Upper bound on s

There is no (152, 260, 5493)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 464135 258428 114755 105176 993268 047330 067958 474663 149266 779950 733176 539303 258904 325607 945534 682732 293331 618356 586477 720777 323506 325257 653504 518715 821981 682736 > 4²⁶⁰ [i]