MinT - Best Known (240−93, 240, s)-Nets in Base 4

Best Known (240−93, 240, s)-Nets in Base 4

(240−93, 240, 138)-Net over F₄ — Constructive and digital

Digital (147, 240, 138)-net over F₄, using

13 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]

(240−93, 240, 146)-Net in Base 4 — Constructive

(147, 240, 146)-net in base 4, using

1 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]

(240−93, 240, 381)-Net over F₄ — Digital

Digital (147, 240, 381)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−93, 240, 8019)-Net in Base 4 — Upper bound on s

There is no (147, 240, 8020)-net in base 4, because

1 times m-reduction [i] would yield (147, 239, 8020)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 781413 109670 725773 181612 592983 179707 597004 582352 734098 167166 368667 323726 791802 999053 963203 946457 394566 876037 349846 273293 692849 253378 218687 096936 > 4²³⁹ [i]