MinT - Best Known (197−73, 197, s)-Nets in Base 4

Best Known (197−73, 197, s)-Nets in Base 4

(197−73, 197, 144)-Net over F₄ — Constructive and digital

Digital (124, 197, 144)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 39, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, 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]

(197−73, 197, 152)-Net in Base 4 — Constructive

(124, 197, 152)-net in base 4, using

1 times m-reduction [i] based on (124, 198, 152)-net in base 4, using
- trace code for nets [i] based on (25, 99, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (25, 100, 76)-net in base 16, using
    - base change [i] based on digital (5, 80, 76)-net over F₃₂, using
      - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(197−73, 197, 374)-Net over F₄ — Digital

Digital (124, 197, 374)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−73, 197, 8996)-Net in Base 4 — Upper bound on s

There is no (124, 197, 8997)-net in base 4, because

1 times m-reduction [i] would yield (124, 196, 8997)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10103 316624 870485 701307 216939 058711 579337 970156 249566 422287 476057 006495 214729 898197 059007 365672 197234 862272 442887 187417 > 4¹⁹⁶ [i]