MinT - Best Known (170−58, 170, s)-Nets in Base 4

Best Known (170−58, 170, s)-Nets in Base 4

(170−58, 170, 160)-Net over F₄ — Constructive and digital

Digital (112, 170, 160)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (13, 42, 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 (70, 128, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 64, 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]

(170−58, 170, 208)-Net in Base 4 — Constructive

(112, 170, 208)-net in base 4, using

t-expansion [i] based on (111, 170, 208)-net in base 4, using
- trace code for nets [i] based on (26, 85, 104)-net in base 16, using
  - base change [i] based on digital (9, 68, 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]

(170−58, 170, 439)-Net over F₄ — Digital

Digital (112, 170, 439)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(170−58, 170, 13138)-Net in Base 4 — Upper bound on s

There is no (112, 170, 13139)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 243543 713546 311156 130313 294417 420104 223628 233797 058521 786822 686908 699988 712613 289695 728895 170340 448736 > 4¹⁷⁰ [i]