MinT - Best Known (234−89, 234, s)-Nets in Base 4

Best Known (234−89, 234, s)-Nets in Base 4

(234−89, 234, 138)-Net over F₄ — Constructive and digital

Digital (145, 234, 138)-net over F₄, using

13 times m-reduction [i] based on digital (145, 247, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 72, 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, 175, 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]

(234−89, 234, 146)-Net in Base 4 — Constructive

(145, 234, 146)-net in base 4, using

1 times m-reduction [i] based on (145, 235, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 72, 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, 163, 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]

(234−89, 234, 397)-Net over F₄ — Digital

Digital (145, 234, 397)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−89, 234, 8835)-Net in Base 4 — Upper bound on s

There is no (145, 234, 8836)-net in base 4, because

1 times m-reduction [i] would yield (145, 233, 8836)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 191 390737 811398 320701 724848 770425 718545 355588 930253 114600 344210 961569 031238 843324 793489 740546 214189 003927 783251 499652 280257 188856 064704 239840 > 4²³³ [i]