MinT - Best Known (209−121, 209, s)-Nets in Base 4

Best Known (209−121, 209, s)-Nets in Base 4

(209−121, 209, 104)-Net over F₄ — Constructive and digital

Digital (88, 209, 104)-net over F₄, using

t-expansion [i] based on digital (73, 209, 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]

(209−121, 209, 129)-Net over F₄ — Digital

Digital (88, 209, 129)-net over F₄, using

t-expansion [i] based on digital (81, 209, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(209−121, 209, 896)-Net in Base 4 — Upper bound on s

There is no (88, 209, 897)-net in base 4, because

1 times m-reduction [i] would yield (88, 208, 897)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 176927 857862 815053 404188 083832 980780 732731 028536 978078 625047 206772 537704 296193 481183 387043 711051 319366 958169 720616 683504 025696 > 4²⁰⁸ [i]