MinT - Best Known (131, 131+73, s)-Nets in Base 4

Best Known (131, 131+73, s)-Nets in Base 4

(131, 131+73, 157)-Net over F₄ — Constructive and digital

Digital (131, 204, 157)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 46, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (85, 158, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 79, 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]

(131, 131+73, 196)-Net in Base 4 — Constructive

(131, 204, 196)-net in base 4, using

2 times m-reduction [i] based on (131, 206, 196)-net in base 4, using
- trace code for nets [i] based on (28, 103, 98)-net in base 16, using
  - 2 times m-reduction [i] based on (28, 105, 98)-net in base 16, using
    - base change [i] based on digital (7, 84, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(131, 131+73, 436)-Net over F₄ — Digital

Digital (131, 204, 436)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+73, 11789)-Net in Base 4 — Upper bound on s

There is no (131, 204, 11790)-net in base 4, because

1 times m-reduction [i] would yield (131, 203, 11790)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 165 714486 388961 584849 844310 653523 566064 380940 983022 183837 671512 466747 951650 762812 203285 346170 215874 542375 561869 103816 335856 > 4²⁰³ [i]