MinT - Best Known (165, 260, s)-Nets in Base 4

Best Known (165, 260, s)-Nets in Base 4

(165, 260, 200)-Net over F₄ — Constructive and digital

Digital (165, 260, 200)-net over F₄, using

t-expansion [i] based on digital (161, 260, 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]

(165, 260, 208)-Net in Base 4 — Constructive

(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]

(165, 260, 501)-Net over F₄ — Digital

Digital (165, 260, 501)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 260, 12687)-Net in Base 4 — Upper bound on s

There is no (165, 260, 12688)-net in base 4, because

1 times m-reduction [i] would yield (165, 259, 12688)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 858121 644802 042442 829284 037195 008367 710832 832076 672774 817818 116309 808353 189503 035844 321326 765275 559396 317674 348919 480138 402882 311681 269539 605840 283998 661180 > 4²⁵⁹ [i]