MinT - Best Known (135, 135+72, s)-Nets in Base 4

Best Known (135, 135+72, s)-Nets in Base 4

(135, 135+72, 163)-Net over F₄ — Constructive and digital

Digital (135, 207, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 51, 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 (84, 156, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 78, 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]

(135, 135+72, 208)-Net in Base 4 — Constructive

(135, 207, 208)-net in base 4, using

3 times m-reduction [i] based on (135, 210, 208)-net in base 4, using
- trace code for nets [i] based on (30, 105, 104)-net in base 16, using
  - base change [i] based on digital (9, 84, 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]

(135, 135+72, 487)-Net over F₄ — Digital

Digital (135, 207, 487)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+72, 13757)-Net in Base 4 — Upper bound on s

There is no (135, 207, 13758)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42391 765249 225657 472523 298349 355590 215569 352572 346026 173085 366874 702037 840402 516568 880617 931275 399992 362733 569314 559632 487105 > 4²⁰⁷ [i]