MinT - Best Known (239−91, 239, s)-Nets in Base 4

Best Known (239−91, 239, s)-Nets in Base 4

(239−91, 239, 138)-Net over F₄ — Constructive and digital

Digital (148, 239, 138)-net over F₄, using

17 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]

(239−91, 239, 147)-Net in Base 4 — Constructive

(148, 239, 147)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (30, 75, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
2. digital (73, 164, 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]

(239−91, 239, 403)-Net over F₄ — Digital

Digital (148, 239, 403)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−91, 239, 8943)-Net in Base 4 — Upper bound on s

There is no (148, 239, 8944)-net in base 4, because

1 times m-reduction [i] would yield (148, 238, 8944)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195942 739552 858003 184922 990751 001524 901925 373186 296684 693380 709717 516677 514608 996907 298486 252621 829160 654080 102740 003186 084837 058971 200967 570405 > 4²³⁸ [i]