MinT - Best Known (183−37, 183, s)-Nets in Base 4

Best Known (183−37, 183, s)-Nets in Base 4

(183−37, 183, 1062)-Net over F₄ — Constructive and digital

Digital (146, 183, 1062)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 31, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (115, 152, 1032)-net over F₄, using
  - trace code for nets [i] based on digital (1, 38, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(183−37, 183, 5489)-Net over F₄ — Digital

Digital (146, 183, 5489)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(183−37, 183, 3079656)-Net in Base 4 — Upper bound on s

There is no (146, 183, 3079657)-net in base 4, because

1 times m-reduction [i] would yield (146, 182, 3079657)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 37 576874 676915 836710 864675 973225 535654 245450 303285 738216 185345 872435 269354 889820 545349 621096 311872 871341 021944 > 4¹⁸² [i]