MinT - Best Known (132, 132+73, s)-Nets in Base 4

Best Known (132, 132+73, s)-Nets in Base 4

(132, 132+73, 157)-Net over F₄ — Constructive and digital

Digital (132, 205, 157)-net over F₄, using

4¹ times duplication [i] based on 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]

(132, 132+73, 196)-Net in Base 4 — Constructive

(132, 205, 196)-net in base 4, using

3 times m-reduction [i] based on (132, 208, 196)-net in base 4, using
- trace code for nets [i] based on (28, 104, 98)-net in base 16, using
  - 1 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]

(132, 132+73, 445)-Net over F₄ — Digital

Digital (132, 205, 445)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(132, 132+73, 12253)-Net in Base 4 — Upper bound on s

There is no (132, 205, 12254)-net in base 4, because

1 times m-reduction [i] would yield (132, 204, 12254)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 662 814722 744409 844543 816464 434542 021692 485780 877615 903416 299017 763963 899069 231043 457490 825629 776523 286009 408722 070608 223562 > 4²⁰⁴ [i]