MinT - Best Known (238−84, 238, s)-Nets in Base 4

Best Known (238−84, 238, s)-Nets in Base 4

(238−84, 238, 163)-Net over F₄ — Constructive and digital

Digital (154, 238, 163)-net over F₄, using

1 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]

(238−84, 238, 240)-Net in Base 4 — Constructive

(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]

(238−84, 238, 524)-Net over F₄ — Digital

Digital (154, 238, 524)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(238−84, 238, 14167)-Net in Base 4 — Upper bound on s

There is no (154, 238, 14168)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195296 654676 423823 151808 423769 757181 952745 542899 106983 781285 153553 625395 104080 142426 887338 316301 188263 084735 769627 530146 947557 068050 583185 858412 > 4²³⁸ [i]