MinT - Best Known (237−82, 237, s)-Nets in Base 4

Best Known (237−82, 237, s)-Nets in Base 4

(237−82, 237, 163)-Net over F₄ — Constructive and digital

Digital (155, 237, 163)-net over F₄, using

t-expansion [i] based on digital (154, 237, 163)-net over F₄, using
- 2 times m-reduction [i] based on digital (154, 239, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 57, 33)-net over F₄, using
      - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
    2. digital (97, 182, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 91, 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]

(237−82, 237, 240)-Net in Base 4 — Constructive

(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]

(237−82, 237, 559)-Net over F₄ — Digital

Digital (155, 237, 559)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−82, 237, 16221)-Net in Base 4 — Upper bound on s

There is no (155, 237, 16222)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48832 557434 038980 115894 338251 057373 194249 579274 403962 038471 345609 778107 141203 599752 282654 582554 739510 607127 793557 252062 815770 408685 345212 390246 > 4²³⁷ [i]