MinT - Best Known (259−149, 259, s)-Nets in Base 4

Best Known (259−149, 259, s)-Nets in Base 4

(259−149, 259, 130)-Net over F₄ — Constructive and digital

Digital (110, 259, 130)-net over F₄, using

t-expansion [i] based on digital (105, 259, 130)-net over F₄, using
- net from sequence [i] based on digital (105, 129)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 105 and N(F) ≥ 130, using
    - T₇ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(259−149, 259, 165)-Net over F₄ — Digital

Digital (110, 259, 165)-net over F₄, using

t-expansion [i] based on digital (109, 259, 165)-net over F₄, using
- net from sequence [i] based on digital (109, 164)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 109 and N(F) ≥ 165, using
    - Drinfeld modules of rank 1 [i]

(259−149, 259, 1128)-Net in Base 4 — Upper bound on s

There is no (110, 259, 1129)-net in base 4, because

1 times m-reduction [i] would yield (110, 258, 1129)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 224291 644066 746160 873151 412612 723895 627836 338209 408649 657449 758013 462557 038137 661338 964485 068633 502348 719330 755161 355469 620888 364353 039000 835683 500511 851264 > 4²⁵⁸ [i]