MinT - Best Known (200−75, 200, s)-Nets in Base 4

Best Known (200−75, 200, s)-Nets in Base 4

(200−75, 200, 139)-Net over F₄ — Constructive and digital

Digital (125, 200, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 38, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (87, 162, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 81, 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]

(200−75, 200, 152)-Net in Base 4 — Constructive

(125, 200, 152)-net in base 4, using

trace code for nets [i] based on (25, 100, 76)-net in base 16, using
- base change [i] based on digital (5, 80, 76)-net over F₃₂, using
  - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]

(200−75, 200, 364)-Net over F₄ — Digital

Digital (125, 200, 364)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(200−75, 200, 8420)-Net in Base 4 — Upper bound on s

There is no (125, 200, 8421)-net in base 4, because

1 times m-reduction [i] would yield (125, 199, 8421)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 646999 349795 376228 618870 151280 068433 256199 478704 937708 983354 869347 412548 660617 384450 864498 282219 903555 393435 213306 839772 > 4¹⁹⁹ [i]