MinT - Best Known (248−131, 248, s)-Nets in Base 4

Best Known (248−131, 248, s)-Nets in Base 4

(248−131, 248, 130)-Net over F₄ — Constructive and digital

Digital (117, 248, 130)-net over F₄, using

t-expansion [i] based on digital (105, 248, 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]

(248−131, 248, 168)-Net over F₄ — Digital

Digital (117, 248, 168)-net over F₄, using

t-expansion [i] based on digital (115, 248, 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]

(248−131, 248, 1566)-Net in Base 4 — Upper bound on s

There is no (117, 248, 1567)-net in base 4, because

1 times m-reduction [i] would yield (117, 247, 1567)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 51881 858279 464607 351449 028565 485081 672911 631020 596044 283459 873741 324835 521666 913565 521935 911197 697785 918067 928269 441151 825546 193290 810541 397568 415216 > 4²⁴⁷ [i]