MinT - Best Known (146, 239, s)-Nets in Base 4

Best Known (146, 239, s)-Nets in Base 4

(146, 239, 138)-Net over F₄ — Constructive and digital

Digital (146, 239, 138)-net over F₄, using

11 times m-reduction [i] based on digital (146, 250, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 73, 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, 177, 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]

(146, 239, 146)-Net in Base 4 — Constructive

(146, 239, 146)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (27, 73, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [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, 166, 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]

(146, 239, 375)-Net over F₄ — Digital

Digital (146, 239, 375)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(146, 239, 7780)-Net in Base 4 — Upper bound on s

There is no (146, 239, 7781)-net in base 4, because

1 times m-reduction [i] would yield (146, 238, 7781)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195595 715932 806519 608651 267898 444131 535418 190661 829366 826991 233201 278014 500589 710504 966351 528134 055318 241473 913871 739393 179747 430472 813091 303072 > 4²³⁸ [i]