MinT - Best Known (217−88, 217, s)-Nets in Base 4

Best Known (217−88, 217, s)-Nets in Base 4

(217−88, 217, 132)-Net over F₄ — Constructive and digital

Digital (129, 217, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 56, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 161, 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]

(217−88, 217, 300)-Net over F₄ — Digital

Digital (129, 217, 300)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−88, 217, 5322)-Net in Base 4 — Upper bound on s

There is no (129, 217, 5323)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44533 135767 673356 990503 682353 749125 816470 212916 756623 715004 132699 816499 022000 554485 947837 932781 263827 348359 012107 106188 942670 922160 > 4²¹⁷ [i]