MinT - Best Known (154, 154+79, s)-Nets in Base 4

Best Known (154, 154+79, s)-Nets in Base 4

(154, 154+79, 164)-Net over F₄ — Constructive and digital

Digital (154, 233, 164)-net over F₄, using

2 times m-reduction [i] based on digital (154, 235, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 61, 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 (93, 174, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 87, 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]

(154, 154+79, 240)-Net in Base 4 — Constructive

(154, 233, 240)-net in base 4, using

5 times m-reduction [i] based on (154, 238, 240)-net in base 4, using
- trace code for nets [i] based on (35, 119, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (35, 120, 120)-net in base 16, using
    - base change [i] based on digital (11, 96, 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]

(154, 154+79, 591)-Net over F₄ — Digital

Digital (154, 233, 591)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(154, 154+79, 19546)-Net in Base 4 — Upper bound on s

There is no (154, 233, 19547)-net in base 4, because

1 times m-reduction [i] would yield (154, 232, 19547)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 638301 839777 863424 625198 096056 489547 079628 268295 556541 767279 172223 260898 465150 185201 163445 803140 091132 733424 046752 418497 972180 651422 386688 > 4²³² [i]