MinT - Best Known (237−90, 237, s)-Nets in Base 4

Best Known (237−90, 237, s)-Nets in Base 4

(237−90, 237, 138)-Net over F₄ — Constructive and digital

Digital (147, 237, 138)-net over F₄, using

16 times m-reduction [i] based on digital (147, 253, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 74, 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, 179, 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]

(237−90, 237, 146)-Net in Base 4 — Constructive

(147, 237, 146)-net in base 4, using

4 times m-reduction [i] based on (147, 241, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 74, 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, 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]

(237−90, 237, 404)-Net over F₄ — Digital

Digital (147, 237, 404)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−90, 237, 8670)-Net in Base 4 — Upper bound on s

There is no (147, 237, 8671)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48847 089451 847039 425543 879796 956126 830045 869297 267730 725857 243672 186073 162389 860484 637130 488804 962561 539278 037537 197616 004169 861105 352313 933612 > 4²³⁷ [i]