MinT - Best Known (253−98, 253, s)-Nets in Base 4

Best Known (253−98, 253, s)-Nets in Base 4

(253−98, 253, 160)-Net over F₄ — Constructive and digital

Digital (155, 253, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 82, 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, 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]

(253−98, 253, 400)-Net over F₄ — Digital

Digital (155, 253, 400)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−98, 253, 8140)-Net in Base 4 — Upper bound on s

There is no (155, 253, 8141)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 210 387272 512880 818269 803089 813371 144824 809841 340223 741989 500545 660242 023420 100632 105531 386161 580196 741573 628496 959676 907743 008547 544151 486508 393804 878496 > 4²⁵³ [i]