MinT - Best Known (167−56, 167, s)-Nets in Base 4

Best Known (167−56, 167, s)-Nets in Base 4

(167−56, 167, 163)-Net over F₄ — Constructive and digital

Digital (111, 167, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 43, 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 (68, 124, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 62, 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]

(167−56, 167, 208)-Net in Base 4 — Constructive

(111, 167, 208)-net in base 4, using

3 times m-reduction [i] based on (111, 170, 208)-net in base 4, using
- trace code for nets [i] based on (26, 85, 104)-net in base 16, using
  - base change [i] based on digital (9, 68, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(167−56, 167, 461)-Net over F₄ — Digital

Digital (111, 167, 461)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−56, 167, 14657)-Net in Base 4 — Upper bound on s

There is no (111, 167, 14658)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 35015 701536 317602 228266 595367 794572 748517 481300 254908 485325 267364 620834 036750 230291 489902 003645 968720 > 4¹⁶⁷ [i]