MinT - Best Known (257−105, 257, s)-Nets in Base 4

Best Known (257−105, 257, s)-Nets in Base 4

(257−105, 257, 138)-Net over F₄ — Constructive and digital

Digital (152, 257, 138)-net over F₄, using

t-expansion [i] based on digital (149, 257, 138)-net over F₄, using
- 2 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]

(257−105, 257, 146)-Net in Base 4 — Constructive

(152, 257, 146)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (27, 79, 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, 178, 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]

(257−105, 257, 341)-Net over F₄ — Digital

Digital (152, 257, 341)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−105, 257, 6162)-Net in Base 4 — Upper bound on s

There is no (152, 257, 6163)-net in base 4, because

1 times m-reduction [i] would yield (152, 256, 6163)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13454 029075 639714 688728 213023 553318 776180 788973 637997 937507 318556 907359 859268 382765 996263 279860 084008 119296 175910 461900 529771 267918 510118 038136 668038 361600 > 4²⁵⁶ [i]