MinT - Best Known (157−53, 157, s)-Nets in Base 4

Best Known (157−53, 157, s)-Nets in Base 4

(157−53, 157, 160)-Net over F₄ — Constructive and digital

Digital (104, 157, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 39, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
2. digital (65, 118, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 59, 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]

(157−53, 157, 208)-Net in Base 4 — Constructive

(104, 157, 208)-net in base 4, using

1 times m-reduction [i] based on (104, 158, 208)-net in base 4, using
- trace code for nets [i] based on (25, 79, 104)-net in base 16, using
  - 1 times m-reduction [i] based on (25, 80, 104)-net in base 16, using
    - base change [i] based on digital (9, 64, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(157−53, 157, 427)-Net over F₄ — Digital

Digital (104, 157, 427)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(157−53, 157, 14384)-Net in Base 4 — Upper bound on s

There is no (104, 157, 14385)-net in base 4, because

1 times m-reduction [i] would yield (104, 156, 14385)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8351 866058 189630 787651 461195 988965 310057 558755 706864 369904 800845 121760 023525 059005 602283 006384 > 4¹⁵⁶ [i]