MinT - Best Known (137, 137+73, s)-Nets in Base 4

Best Known (137, 137+73, s)-Nets in Base 4

(137, 137+73, 163)-Net over F₄ — Constructive and digital

Digital (137, 210, 163)-net over F₄, using

4¹ times duplication [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+73, 240)-Net in Base 4 — Constructive

(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+73, 494)-Net over F₄ — Digital

Digital (137, 210, 494)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+73, 14860)-Net in Base 4 — Upper bound on s

There is no (137, 210, 14861)-net in base 4, because

1 times m-reduction [i] would yield (137, 209, 14861)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 676925 089494 325928 034219 288838 328225 674121 936794 818434 070724 469447 124838 607114 153532 896978 082121 360099 348719 049025 396731 796752 > 4²⁰⁹ [i]