MinT - Best Known (233−99, 233, s)-Nets in Base 4

Best Known (233−99, 233, s)-Nets in Base 4

(233−99, 233, 132)-Net over F₄ — Constructive and digital

Digital (134, 233, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 61, 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, 172, 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]

(233−99, 233, 275)-Net over F₄ — Digital

Digital (134, 233, 275)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(233−99, 233, 4475)-Net in Base 4 — Upper bound on s

There is no (134, 233, 4476)-net in base 4, because

1 times m-reduction [i] would yield (134, 232, 4476)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 47 720336 487797 692006 101485 675942 543225 690131 843893 401928 964203 556224 743498 794438 656973 432315 245090 362905 234913 260815 227386 199594 932431 853261 > 4²³² [i]