MinT - Best Known (223−92, 223, s)-Nets in Base 4

Best Known (223−92, 223, s)-Nets in Base 4

(223−92, 223, 132)-Net over F₄ — Constructive and digital

Digital (131, 223, 132)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 58, 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, 165, 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]

(223−92, 223, 291)-Net over F₄ — Digital

Digital (131, 223, 291)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−92, 223, 4937)-Net in Base 4 — Upper bound on s

There is no (131, 223, 4938)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 182 976578 958703 995752 611732 097667 107437 051654 055076 524765 243401 098877 765528 729942 532519 068934 781370 671038 798340 629893 191717 928490 101600 > 4²²³ [i]