MinT - Best Known (197−68, 197, s)-Nets in Base 4

Best Known (197−68, 197, s)-Nets in Base 4

(197−68, 197, 163)-Net over F₄ — Constructive and digital

Digital (129, 197, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 49, 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 (80, 148, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 74, 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]

(197−68, 197, 208)-Net in Base 4 — Constructive

(129, 197, 208)-net in base 4, using

3 times m-reduction [i] based on (129, 200, 208)-net in base 4, using
- trace code for nets [i] based on (29, 100, 104)-net in base 16, using
  - base change [i] based on digital (9, 80, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(197−68, 197, 478)-Net over F₄ — Digital

Digital (129, 197, 478)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−68, 197, 13864)-Net in Base 4 — Upper bound on s

There is no (129, 197, 13865)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40441 140761 231136 881656 991217 931075 007815 597128 839848 194561 610962 705592 125016 025340 161967 642526 025894 632526 516632 510300 > 4¹⁹⁷ [i]