MinT - Best Known (257−91, 257, s)-Nets in Base 4

Best Known (257−91, 257, s)-Nets in Base 4

(257−91, 257, 200)-Net over F₄ — Constructive and digital

Digital (166, 257, 200)-net over F₄, using

t-expansion [i] based on digital (161, 257, 200)-net over F₄, using
- net from sequence [i] based on digital (161, 199)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 161 and N(F) ≥ 200, using
    - F₇ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(257−91, 257, 240)-Net in Base 4 — Constructive

(166, 257, 240)-net in base 4, using

1 times m-reduction [i] based on (166, 258, 240)-net in base 4, using
- trace code for nets [i] based on (37, 129, 120)-net in base 16, using
  - 1 times m-reduction [i] based on (37, 130, 120)-net in base 16, using
    - base change [i] based on digital (11, 104, 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]

(257−91, 257, 554)-Net over F₄ — Digital

Digital (166, 257, 554)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−91, 257, 15598)-Net in Base 4 — Upper bound on s

There is no (166, 257, 15599)-net in base 4, because

1 times m-reduction [i] would yield (166, 256, 15599)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13435 614071 663421 065557 612811 395298 053957 035367 718979 726957 469179 464885 786735 279180 113309 029847 429960 844309 038944 423046 733459 012105 899459 694875 903685 921458 > 4²⁵⁶ [i]