MinT - Best Known (128, 128+74, s)-Nets in Base 4

Best Known (128, 128+74, s)-Nets in Base 4

(128, 128+74, 147)-Net over F₄ — Constructive and digital

Digital (128, 202, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 42, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (86, 160, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 80, 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]

(128, 128+74, 152)-Net in Base 4 — Constructive

(128, 202, 152)-net in base 4, using

2 times m-reduction [i] based on (128, 204, 152)-net in base 4, using
- trace code for nets [i] based on (26, 102, 76)-net in base 16, using
  - 3 times m-reduction [i] based on (26, 105, 76)-net in base 16, using
    - base change [i] based on digital (5, 84, 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]

(128, 128+74, 398)-Net over F₄ — Digital

Digital (128, 202, 398)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+74, 9425)-Net in Base 4 — Upper bound on s

There is no (128, 202, 9426)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 41 345553 706402 176973 530922 413502 424132 238845 237873 504011 962949 226724 413958 209416 428281 268887 555356 389442 893586 091914 452520 > 4²⁰² [i]