MinT - Best Known (257−101, 257, s)-Nets in Base 4

Best Known (257−101, 257, s)-Nets in Base 4

(257−101, 257, 160)-Net over F₄ — Constructive and digital

Digital (156, 257, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 83, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 174, 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]

(257−101, 257, 387)-Net over F₄ — Digital

Digital (156, 257, 387)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−101, 257, 7812)-Net in Base 4 — Upper bound on s

There is no (156, 257, 7813)-net in base 4, because

1 times m-reduction [i] would yield (156, 256, 7813)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13420 411930 945160 730070 600236 013174 105465 604966 085011 939341 220804 471691 297252 622197 777021 147926 475499 836019 115123 029096 371351 320425 290645 743448 330432 107248 > 4²⁵⁶ [i]