MinT - Best Known (166, 166+93, s)-Nets in Base 4

Best Known (166, 166+93, s)-Nets in Base 4

(166, 166+93, 200)-Net over F₄ — Constructive and digital

Digital (166, 259, 200)-net over F₄, using

t-expansion [i] based on digital (161, 259, 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]

(166, 166+93, 208)-Net in Base 4 — Constructive

(166, 259, 208)-net in base 4, using

t-expansion [i] based on (165, 259, 208)-net in base 4, using
- 1 times m-reduction [i] based on (165, 260, 208)-net in base 4, using
  - trace code for nets [i] based on (35, 130, 104)-net in base 16, using
    - base change [i] based on digital (9, 104, 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]

(166, 166+93, 531)-Net over F₄ — Digital

Digital (166, 259, 531)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(166, 166+93, 14247)-Net in Base 4 — Upper bound on s

There is no (166, 259, 14248)-net in base 4, because

1 times m-reduction [i] would yield (166, 258, 14248)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 215085 475588 780393 436622 919427 988944 211373 120819 238216 263885 040553 026346 274469 981849 602162 295122 502895 787230 675224 713022 399511 558288 566423 413230 147456 623440 > 4²⁵⁸ [i]