MinT - Best Known (151, 151+106, s)-Nets in Base 4

Best Known (151, 151+106, s)-Nets in Base 4

(151, 151+106, 138)-Net over F₄ — Constructive and digital

Digital (151, 257, 138)-net over F₄, using

t-expansion [i] based on digital (149, 257, 138)-net over F₄, using
- 2 times m-reduction [i] based on digital (149, 259, 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, 183, 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]

(151, 151+106, 139)-Net in Base 4 — Constructive

(151, 257, 139)-net in base 4, using

2 times m-reduction [i] based on (151, 259, 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, 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]

(151, 151+106, 331)-Net over F₄ — Digital

Digital (151, 257, 331)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(151, 151+106, 5659)-Net in Base 4 — Upper bound on s

There is no (151, 257, 5660)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53970 629580 707500 755338 746196 666783 092664 446428 581870 119682 385886 382845 394746 139541 538973 080101 808898 183556 075446 190078 004930 777188 227212 440104 673534 764320 > 4²⁵⁷ [i]