MinT - Best Known (255−115, 255, s)-Nets in Base 4

Best Known (255−115, 255, s)-Nets in Base 4

(255−115, 255, 131)-Net over F₄ — Constructive and digital

Digital (140, 255, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 67, 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 (73, 188, 104)-net over F₄, using
  - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
      - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(255−115, 255, 247)-Net over F₄ — Digital

Digital (140, 255, 247)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−115, 255, 3499)-Net in Base 4 — Upper bound on s

There is no (140, 255, 3500)-net in base 4, because

1 times m-reduction [i] would yield (140, 254, 3500)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 848 409278 263834 424565 590691 839414 451950 850485 403539 150169 673989 905648 878971 580624 750371 870245 079181 694970 640889 468622 171444 704696 925836 287124 657265 029896 > 4²⁵⁴ [i]