MinT - Best Known (228−96, 228, s)-Nets in Base 4

Best Known (228−96, 228, s)-Nets in Base 4

(228−96, 228, 131)-Net over F₄ — Constructive and digital

Digital (132, 228, 131)-net over F₄, using

2 times m-reduction [i] based on digital (132, 230, 131)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (10, 59, 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, 171, 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]

(228−96, 228, 278)-Net over F₄ — Digital

Digital (132, 228, 278)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−96, 228, 4483)-Net in Base 4 — Upper bound on s

There is no (132, 228, 4484)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 186245 725375 606367 432079 288726 085337 126572 280346 806750 100055 946769 330631 810043 465358 495056 535311 456168 381418 830705 671887 941421 264160 924888 > 4²²⁸ [i]