MinT - Best Known (237−81, 237, s)-Nets in Base 4

Best Known (237−81, 237, s)-Nets in Base 4

(237−81, 237, 164)-Net over F₄ — Constructive and digital

Digital (156, 237, 164)-net over F₄, using

1 times m-reduction [i] based on digital (156, 238, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 62, 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 (94, 176, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 88, 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]

(237−81, 237, 240)-Net in Base 4 — Constructive

(156, 237, 240)-net in base 4, using

t-expansion [i] based on (155, 237, 240)-net in base 4, using
- 3 times m-reduction [i] based on (155, 240, 240)-net in base 4, using
  - trace code for nets [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]

(237−81, 237, 584)-Net over F₄ — Digital

Digital (156, 237, 584)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−81, 237, 18709)-Net in Base 4 — Upper bound on s

There is no (156, 237, 18710)-net in base 4, because

1 times m-reduction [i] would yield (156, 236, 18710)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12206 592189 008764 287306 418431 111635 033405 485975 693479 533789 992135 931366 097447 120913 023356 099084 157643 347549 122205 672618 597452 341220 825437 011184 > 4²³⁶ [i]