MinT - Best Known (191−63, 191, s)-Nets in Base 4

Best Known (191−63, 191, s)-Nets in Base 4

(191−63, 191, 195)-Net over F₄ — Constructive and digital

Digital (128, 191, 195)-net over F₄, using

1 times m-reduction [i] based on digital (128, 192, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 64, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(191−63, 191, 240)-Net in Base 4 — Constructive

(128, 191, 240)-net in base 4, using

3 times m-reduction [i] based on (128, 194, 240)-net in base 4, using
- trace code for nets [i] based on (31, 97, 120)-net in base 16, using
  - 3 times m-reduction [i] based on (31, 100, 120)-net in base 16, using
    - base change [i] based on digital (11, 80, 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]

(191−63, 191, 549)-Net over F₄ — Digital

Digital (128, 191, 549)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(191−63, 191, 20249)-Net in Base 4 — Upper bound on s

There is no (128, 191, 20250)-net in base 4, because

1 times m-reduction [i] would yield (128, 190, 20250)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 464547 223796 672394 910087 454452 175012 762316 471462 800598 041724 738428 115779 595861 180609 349075 654319 880022 460819 363456 > 4¹⁹⁰ [i]