MinT - Best Known (245−85, 245, s)-Nets in Base 4

Best Known (245−85, 245, s)-Nets in Base 4

(245−85, 245, 164)-Net over F₄ — Constructive and digital

Digital (160, 245, 164)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (21, 63, 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 (97, 182, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 91, 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]

(245−85, 245, 240)-Net in Base 4 — Constructive

(160, 245, 240)-net in base 4, using

3 times m-reduction [i] based on (160, 248, 240)-net in base 4, using
- trace code for nets [i] based on (36, 124, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (36, 125, 120)-net in base 16, using
    - base change [i] based on digital (11, 100, 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]

(245−85, 245, 571)-Net over F₄ — Digital

Digital (160, 245, 571)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−85, 245, 17278)-Net in Base 4 — Upper bound on s

There is no (160, 245, 17279)-net in base 4, because

1 times m-reduction [i] would yield (160, 244, 17279)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 800 850316 062796 599068 306315 390707 452622 323149 920820 222555 508448 339366 039428 937321 338344 637216 744964 376173 650120 918694 482202 183838 084873 735524 376695 > 4²⁴⁴ [i]