MinT - Best Known (254−113, 254, s)-Nets in Base 4

Best Known (254−113, 254, s)-Nets in Base 4

(254−113, 254, 132)-Net over F₄ — Constructive and digital

Digital (141, 254, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 68, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 186, 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]

(254−113, 254, 257)-Net over F₄ — Digital

Digital (141, 254, 257)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−113, 254, 3751)-Net in Base 4 — Upper bound on s

There is no (141, 254, 3752)-net in base 4, because

1 times m-reduction [i] would yield (141, 253, 3752)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 209 749689 877831 513515 009656 343357 399717 006381 256287 998114 900558 070930 941088 412092 130137 967583 306300 390437 281212 450618 424456 976300 098246 501941 051642 344528 > 4²⁵³ [i]