MinT - Best Known (253−97, 253, s)-Nets in Base 4

Best Known (253−97, 253, s)-Nets in Base 4

(253−97, 253, 160)-Net over F₄ — Constructive and digital

Digital (156, 253, 160)-net over F₄, using

3 times m-reduction [i] based on digital (156, 256, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 83, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [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]

(253−97, 253, 414)-Net over F₄ — Digital

Digital (156, 253, 414)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−97, 253, 9007)-Net in Base 4 — Upper bound on s

There is no (156, 253, 9008)-net in base 4, because

1 times m-reduction [i] would yield (156, 252, 9008)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52 617728 095252 083831 657874 387559 156162 691110 943218 657754 398989 684963 971314 528053 794409 154005 335311 849207 985699 547392 426521 104244 459591 026829 078592 727120 > 4²⁵² [i]