MinT - Best Known (249−99, 249, s)-Nets in Base 4

Best Known (249−99, 249, s)-Nets in Base 4

(249−99, 249, 138)-Net over F₄ — Constructive and digital

Digital (150, 249, 138)-net over F₄, using

t-expansion [i] based on digital (149, 249, 138)-net over F₄, using
- 10 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]

(249−99, 249, 146)-Net in Base 4 — Constructive

(150, 249, 146)-net in base 4, using

1 times m-reduction [i] based on (150, 250, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 77, 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, 173, 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]

(249−99, 249, 362)-Net over F₄ — Digital

Digital (150, 249, 362)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−99, 249, 7061)-Net in Base 4 — Upper bound on s

There is no (150, 249, 7062)-net in base 4, because

1 times m-reduction [i] would yield (150, 248, 7062)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 205696 185042 123552 215066 577038 582050 636700 775596 240630 714105 395620 624210 970202 914290 119982 522652 706661 380755 252718 192626 447519 049554 444553 487636 450340 > 4²⁴⁸ [i]