MinT - Best Known (239−89, 239, s)-Nets in Base 4

Best Known (239−89, 239, s)-Nets in Base 4

(239−89, 239, 160)-Net over F₄ — Constructive and digital

Digital (150, 239, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (33, 77, 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, 162, 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]

(239−89, 239, 435)-Net over F₄ — Digital

Digital (150, 239, 435)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−89, 239, 10348)-Net in Base 4 — Upper bound on s

There is no (150, 239, 10349)-net in base 4, because

1 times m-reduction [i] would yield (150, 238, 10349)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195361 940080 653129 132062 594288 878875 797358 074934 524570 597246 088849 230970 918407 473067 653677 793852 360054 383674 569749 167271 443649 971039 340568 523767 > 4²³⁸ [i]