MinT - Best Known (144, 144+93, s)-Nets in Base 4

Best Known (144, 144+93, s)-Nets in Base 4

(144, 144+93, 138)-Net over F₄ — Constructive and digital

Digital (144, 237, 138)-net over F₄, using

7 times m-reduction [i] based on digital (144, 244, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 71, 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, 173, 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]

(144, 144+93, 139)-Net in Base 4 — Constructive

(144, 237, 139)-net in base 4, using

1 times m-reduction [i] based on (144, 238, 139)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (24, 71, 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, 167, 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]

(144, 144+93, 362)-Net over F₄ — Digital

Digital (144, 237, 362)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(144, 144+93, 7323)-Net in Base 4 — Upper bound on s

There is no (144, 237, 7324)-net in base 4, because

1 times m-reduction [i] would yield (144, 236, 7324)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12250 721486 425003 529747 355420 399666 687116 827167 545933 227364 978983 595149 653262 900114 859203 134983 755595 872317 112615 058400 997709 447124 490385 553040 > 4²³⁶ [i]