MinT - Best Known (253−106, 253, s)-Nets in Base 4

Best Known (253−106, 253, s)-Nets in Base 4

(253−106, 253, 138)-Net over F₄ — Constructive and digital

Digital (147, 253, 138)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 74, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (73, 179, 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]

(253−106, 253, 310)-Net over F₄ — Digital

Digital (147, 253, 310)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−106, 253, 5092)-Net in Base 4 — Upper bound on s

There is no (147, 253, 5093)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 209 885360 055903 785352 736119 209821 163394 085967 140643 913529 898652 640718 840302 379700 623175 837341 840828 496686 462547 805852 359487 992080 552718 575815 392852 086800 > 4²⁵³ [i]