MinT - Best Known (215−72, 215, s)-Nets in Base 4

Best Known (215−72, 215, s)-Nets in Base 4

(215−72, 215, 164)-Net over F₄ — Constructive and digital

Digital (143, 215, 164)-net over F₄, using

t-expansion [i] based on digital (142, 215, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 57, 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 (85, 158, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 79, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(215−72, 215, 240)-Net in Base 4 — Constructive

(143, 215, 240)-net in base 4, using

5 times m-reduction [i] based on (143, 220, 240)-net in base 4, using
- trace code for nets [i] based on (33, 110, 120)-net in base 16, using
  - base change [i] based on digital (11, 88, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(215−72, 215, 576)-Net over F₄ — Digital

Digital (143, 215, 576)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−72, 215, 18731)-Net in Base 4 — Upper bound on s

There is no (143, 215, 18732)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2776 177798 123654 691821 380638 286509 273805 131905 365890 059823 434215 174225 444232 569668 647745 736742 913894 885230 583182 045325 132874 774610 > 4²¹⁵ [i]