MinT - Best Known (253−99, 253, s)-Nets in Base 4

Best Known (253−99, 253, s)-Nets in Base 4

(253−99, 253, 138)-Net over F₄ — Constructive and digital

Digital (154, 253, 138)-net over F₄, using

t-expansion [i] based on digital (149, 253, 138)-net over F₄, using
- 6 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]

(253−99, 253, 147)-Net in Base 4 — Constructive

(154, 253, 147)-net in base 4, using

3 times m-reduction [i] based on (154, 256, 147)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (30, 81, 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, 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]

(253−99, 253, 387)-Net over F₄ — Digital

Digital (154, 253, 387)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−99, 253, 7912)-Net in Base 4 — Upper bound on s

There is no (154, 253, 7913)-net in base 4, because

1 times m-reduction [i] would yield (154, 252, 7913)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 666899 600816 787976 844144 280041 024471 685956 866301 328024 704222 095752 951117 378700 837714 606314 764019 948364 382667 553264 028654 685206 825476 617404 379268 644432 > 4²⁵² [i]