MinT - Best Known (250−99, 250, s)-Nets in Base 4

Best Known (250−99, 250, s)-Nets in Base 4

(250−99, 250, 138)-Net over F₄ — Constructive and digital

Digital (151, 250, 138)-net over F₄, using

t-expansion [i] based on digital (149, 250, 138)-net over F₄, using
- 9 times m-reduction [i] based on digital (149, 259, 138)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (21, 76, 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, 183, 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]

(250−99, 250, 146)-Net in Base 4 — Constructive

(151, 250, 146)-net in base 4, using

3 times m-reduction [i] based on (151, 253, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 78, 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, 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]

(250−99, 250, 368)-Net over F₄ — Digital

Digital (151, 250, 368)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(250−99, 250, 7264)-Net in Base 4 — Upper bound on s

There is no (151, 250, 7265)-net in base 4, because

1 times m-reduction [i] would yield (151, 249, 7265)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 818372 216532 055699 784242 197243 456573 358355 225695 998798 010075 708945 978661 870755 872813 868387 161072 151628 783121 117041 506250 893582 654599 394366 315835 478064 > 4²⁴⁹ [i]