MinT - Best Known (87, 87+48, s)-Nets in Base 4

Best Known (87, 87+48, s)-Nets in Base 4

(87, 87+48, 144)-Net over F₄ — Constructive and digital

Digital (87, 135, 144)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 27, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (60, 108, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 54, 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]

(87, 87+48, 152)-Net in Base 4 — Constructive

(87, 135, 152)-net in base 4, using

1 times m-reduction [i] based on (87, 136, 152)-net in base 4, using
- trace code for nets [i] based on (19, 68, 76)-net in base 16, using
  - 2 times m-reduction [i] based on (19, 70, 76)-net in base 16, using
    - base change [i] based on digital (5, 56, 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]

(87, 87+48, 312)-Net over F₄ — Digital

Digital (87, 135, 312)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(87, 87+48, 7938)-Net in Base 4 — Upper bound on s

There is no (87, 135, 7939)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1897 272184 108877 662646 477410 645559 696897 190945 634474 266239 705208 451711 120937 546464 > 4¹³⁵ [i]