MinT - Best Known (259−102, 259, s)-Nets in Base 4

Best Known (259−102, 259, s)-Nets in Base 4

(259−102, 259, 160)-Net over F₄ — Constructive and digital

Digital (157, 259, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 84, 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, 175, 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]

(259−102, 259, 387)-Net over F₄ — Digital

Digital (157, 259, 387)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−102, 259, 7513)-Net in Base 4 — Upper bound on s

There is no (157, 259, 7514)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 861325 438523 212314 072095 604514 657920 158727 569828 372916 326510 010510 728302 347605 644706 753433 664496 448313 685414 438494 031397 629298 405899 492613 313757 352463 082728 > 4²⁵⁹ [i]