MinT - Best Known (249−101, 249, s)-Nets in Base 4

Best Known (249−101, 249, s)-Nets in Base 4

(249−101, 249, 138)-Net over F₄ — Constructive and digital

Digital (148, 249, 138)-net over F₄, using

7 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]

(249−101, 249, 139)-Net in Base 4 — Constructive

(148, 249, 139)-net in base 4, using

1 times m-reduction [i] based on (148, 250, 139)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (24, 75, 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, 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]

(249−101, 249, 340)-Net over F₄ — Digital

Digital (148, 249, 340)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−101, 249, 6250)-Net in Base 4 — Upper bound on s

There is no (148, 249, 6251)-net in base 4, because

1 times m-reduction [i] would yield (148, 248, 6251)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 205420 733074 765081 642644 353175 351237 483713 612455 746643 627370 428842 815692 846453 264500 311360 637151 654909 116639 203997 024098 202689 988015 153818 077558 705876 > 4²⁴⁸ [i]