MinT - Best Known (152, 239, s)-Nets in Base 4

Best Known (152, 239, s)-Nets in Base 4

(152, 239, 160)-Net over F₄ — Constructive and digital

Digital (152, 239, 160)-net over F₄, using

5 times m-reduction [i] based on digital (152, 244, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 79, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [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]

(152, 239, 196)-Net in Base 4 — Constructive

(152, 239, 196)-net in base 4, using

t-expansion [i] based on (151, 239, 196)-net in base 4, using
- 1 times m-reduction [i] based on (151, 240, 196)-net in base 4, using
  - trace code for nets [i] based on (31, 120, 98)-net in base 16, using
    - base change [i] based on digital (7, 96, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(152, 239, 471)-Net over F₄ — Digital

Digital (152, 239, 471)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(152, 239, 12061)-Net in Base 4 — Upper bound on s

There is no (152, 239, 12062)-net in base 4, because

1 times m-reduction [i] would yield (152, 238, 12062)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 195281 535505 777981 859910 745822 877800 216437 236161 737565 006031 612203 797999 168233 529590 248706 929333 138692 817595 678699 679067 427585 079772 024804 849360 > 4²³⁸ [i]