MinT - Best Known (137, 137+71, s)-Nets in Base 4

Best Known (137, 137+71, s)-Nets in Base 4

(137, 137+71, 163)-Net over F₄ — Constructive and digital

Digital (137, 208, 163)-net over F₄, using

t-expansion [i] based on digital (136, 208, 163)-net over F₄, using
- 1 times m-reduction [i] based on digital (136, 209, 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 (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]

(137, 137+71, 240)-Net in Base 4 — Constructive

(137, 208, 240)-net in base 4, using

2 times m-reduction [i] based on (137, 210, 240)-net in base 4, using
- trace code for nets [i] based on (32, 105, 120)-net in base 16, using
  - base change [i] based on digital (11, 84, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(137, 137+71, 522)-Net over F₄ — Digital

Digital (137, 208, 522)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+71, 16832)-Net in Base 4 — Upper bound on s

There is no (137, 208, 16833)-net in base 4, because

1 times m-reduction [i] would yield (137, 207, 16833)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 42316 710854 638083 132018 507866 870445 926033 102321 278476 345238 863102 718143 535036 701982 620620 734550 350208 381472 138436 679507 149120 > 4²⁰⁷ [i]