MinT - Best Known (209−84, 209, s)-Nets in Base 4

Best Known (209−84, 209, s)-Nets in Base 4

(209−84, 209, 131)-Net over F₄ — Constructive and digital

Digital (125, 209, 131)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 52, 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, 157, 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]

(209−84, 209, 299)-Net over F₄ — Digital

Digital (125, 209, 299)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−84, 209, 5418)-Net in Base 4 — Upper bound on s

There is no (125, 209, 5419)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 678420 578810 775934 700682 432043 188465 005569 161936 855438 838836 193388 959035 688215 520609 290001 858494 513749 829939 705322 931090 832385 > 4²⁰⁹ [i]