MinT - Best Known (127, 127+78, s)-Nets in Base 4

Best Known (127, 127+78, s)-Nets in Base 4

(127, 127+78, 137)-Net over F₄ — Constructive and digital

Digital (127, 205, 137)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 54, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (73, 151, 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]

(127, 127+78, 353)-Net over F₄ — Digital

Digital (127, 205, 353)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+78, 7466)-Net in Base 4 — Upper bound on s

There is no (127, 205, 7467)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2649 455621 625650 650826 473284 367983 507449 222964 749766 243334 890382 072563 080751 092475 396038 579932 820093 257092 721641 894611 205280 > 4²⁰⁵ [i]