MinT - Best Known (125, 125+62, s)-Nets in Base 4

Best Known (125, 125+62, s)-Nets in Base 4

(125, 125+62, 195)-Net over F₄ — Constructive and digital

Digital (125, 187, 195)-net over F₄, using

4¹ times duplication [i] based on digital (124, 186, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 62, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(125, 125+62, 240)-Net in Base 4 — Constructive

(125, 187, 240)-net in base 4, using

3 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+62, 529)-Net over F₄ — Digital

Digital (125, 187, 529)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(125, 125+62, 17704)-Net in Base 4 — Upper bound on s

There is no (125, 187, 17705)-net in base 4, because

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]