MinT - Best Known (241−93, 241, s)-Nets in Base 4

Best Known (241−93, 241, s)-Nets in Base 4

(241−93, 241, 138)-Net over F₄ — Constructive and digital

Digital (148, 241, 138)-net over F₄, using

15 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]

(241−93, 241, 146)-Net in Base 4 — Constructive

(148, 241, 146)-net in base 4, using

3 times m-reduction [i] based on (148, 244, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 75, 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, 169, 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]

(241−93, 241, 388)-Net over F₄ — Digital

Digital (148, 241, 388)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−93, 241, 8266)-Net in Base 4 — Upper bound on s

There is no (148, 241, 8267)-net in base 4, because

1 times m-reduction [i] would yield (148, 240, 8267)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 133805 368381 096588 961382 899133 905548 282048 190466 800564 255840 975468 351167 222539 854948 778683 742768 255982 626077 287172 668081 874562 220390 900855 969552 > 4²⁴⁰ [i]