MinT - Best Known (252−99, 252, s)-Nets in Base 4

Best Known (252−99, 252, s)-Nets in Base 4

(252−99, 252, 138)-Net over F₄ — Constructive and digital

Digital (153, 252, 138)-net over F₄, using

t-expansion [i] based on digital (149, 252, 138)-net over F₄, using
- 7 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]

(252−99, 252, 147)-Net in Base 4 — Constructive

(153, 252, 147)-net in base 4, using

1 times m-reduction [i] based on (153, 253, 147)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (30, 80, 43)-net in base 4, using
    - net from sequence [i] based on (30, 42)-sequence in base 4, using
      - base expansion [i] based on digital (60, 42)-sequence over F₂, using
        t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [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]

(252−99, 252, 381)-Net over F₄ — Digital

Digital (153, 252, 381)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−99, 252, 7690)-Net in Base 4 — Upper bound on s

There is no (153, 252, 7691)-net in base 4, because

1 times m-reduction [i] would yield (153, 251, 7691)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 154703 442332 400595 179310 490399 143989 912041 646453 918000 469999 235507 435997 011593 973356 206482 029799 105754 349960 443471 041295 798334 795322 935646 320636 418880 > 4²⁵¹ [i]