MinT - Best Known (227−76, 227, s)-Nets in Base 4

Best Known (227−76, 227, s)-Nets in Base 4

(227−76, 227, 164)-Net over F₄ — Constructive and digital

Digital (151, 227, 164)-net over F₄, using

3 times m-reduction [i] based on digital (151, 230, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 60, 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 (91, 170, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 85, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(227−76, 227, 240)-Net in Base 4 — Constructive

(151, 227, 240)-net in base 4, using

t-expansion [i] based on (149, 227, 240)-net in base 4, using
- 3 times m-reduction [i] based on (149, 230, 240)-net in base 4, using
  - trace code for nets [i] based on (34, 115, 120)-net in base 16, using
    - base change [i] based on digital (11, 92, 120)-net over F₃₂, using
      - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(227−76, 227, 605)-Net over F₄ — Digital

Digital (151, 227, 605)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(227−76, 227, 19747)-Net in Base 4 — Upper bound on s

There is no (151, 227, 19748)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 46555 942857 593707 968084 510570 084305 313221 865387 607201 122113 817566 107182 962463 818202 013107 690951 171174 017694 809073 418267 826898 833926 033600 > 4²²⁷ [i]