MinT - Best Known (151, 151+105, s)-Nets in Base 4

Best Known (151, 151+105, s)-Nets in Base 4

(151, 151+105, 138)-Net over F₄ — Constructive and digital

Digital (151, 256, 138)-net over F₄, using

t-expansion [i] based on digital (149, 256, 138)-net over F₄, using
- 3 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+105, 140)-Net in Base 4 — Constructive

(151, 256, 140)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (26, 78, 36)-net in base 4, using
  - net from sequence [i] based on (26, 35)-sequence in base 4, using
    - base expansion [i] based on digital (52, 35)-sequence over F₂, using
      - t-expansion [i] based on digital (51, 35)-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 3 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, 178, 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+105, 336)-Net over F₄ — Digital

Digital (151, 256, 336)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(151, 151+105, 5999)-Net in Base 4 — Upper bound on s

There is no (151, 256, 6000)-net in base 4, because

1 times m-reduction [i] would yield (151, 255, 6000)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3370 826649 654323 464245 693387 648832 337645 047252 813210 444688 318322 070837 097735 504605 064193 754078 835374 153441 624645 456797 978060 048849 747390 988201 026594 140596 > 4²⁵⁵ [i]