MinT - Best Known (125, 125+63, s)-Nets in Base 4

Best Known (125, 125+63, s)-Nets in Base 4

(125, 125+63, 163)-Net over F₄ — Constructive and digital

Digital (125, 188, 163)-net over F₄, using

t-expansion [i] based on digital (124, 188, 163)-net over F₄, using
- 1 times m-reduction [i] based on digital (124, 189, 163)-net over F₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (15, 47, 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 (77, 142, 130)-net over F₄, using
      - trace code for nets [i] based on digital (6, 71, 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]

(125, 125+63, 240)-Net in Base 4 — Constructive

(125, 188, 240)-net in base 4, using

2 times m-reduction [i] based on (125, 190, 240)-net in base 4, using
- trace code for nets [i] based on (30, 95, 120)-net in base 16, using
  - base change [i] based on digital (11, 76, 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]

(125, 125+63, 511)-Net over F₄ — Digital

Digital (125, 188, 511)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+63, 17704)-Net in Base 4 — Upper bound on s

There is no (125, 188, 17705)-net in base 4, because

1 times m-reduction [i] would yield (125, 187, 17705)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 38543 441247 770660 059955 439900 137785 044653 879528 130156 347403 565972 388085 625057 153192 537926 215194 465035 282080 768032 > 4¹⁸⁷ [i]