MinT - Best Known (257−104, 257, s)-Nets in Base 4

Best Known (257−104, 257, s)-Nets in Base 4

(257−104, 257, 138)-Net over F₄ — Constructive and digital

Digital (153, 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]

(257−104, 257, 146)-Net in Base 4 — Constructive

(153, 257, 146)-net in base 4, using

2 times m-reduction [i] based on (153, 259, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 80, 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, 179, 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]

(257−104, 257, 352)-Net over F₄ — Digital

Digital (153, 257, 352)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−104, 257, 6330)-Net in Base 4 — Upper bound on s

There is no (153, 257, 6331)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53964 059676 264932 488724 210265 258496 288937 757909 176573 911286 313370 059378 405218 649139 484512 480595 184801 368063 982820 696152 220840 342423 976420 657577 590791 970710 > 4²⁵⁷ [i]