MinT - Best Known (254−133, 254, s)-Nets in Base 4

Best Known (254−133, 254, s)-Nets in Base 4

(254−133, 254, 130)-Net over F₄ — Constructive and digital

Digital (121, 254, 130)-net over F₄, using

t-expansion [i] based on digital (105, 254, 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]

(254−133, 254, 168)-Net over F₄ — Digital

Digital (121, 254, 168)-net over F₄, using

t-expansion [i] based on digital (115, 254, 168)-net over F₄, using
- net from sequence [i] based on digital (115, 167)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 115 and N(F) ≥ 168, using
    - Drinfeld modules of rank 1 [i]

(254−133, 254, 1667)-Net in Base 4 — Upper bound on s

There is no (121, 254, 1668)-net in base 4, because

1 times m-reduction [i] would yield (121, 253, 1668)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 213 724048 763626 653311 360289 507692 761178 811623 722097 197071 239608 805238 812344 186624 160108 663640 314280 038725 439046 789074 532475 577593 174303 856484 020852 987050 > 4²⁵³ [i]