MinT - Best Known (248−137, 248, s)-Nets in Base 4

Best Known (248−137, 248, s)-Nets in Base 4

(248−137, 248, 130)-Net over F₄ — Constructive and digital

Digital (111, 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]

(248−137, 248, 165)-Net over F₄ — Digital

Digital (111, 248, 165)-net over F₄, using

t-expansion [i] based on digital (109, 248, 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]

(248−137, 248, 1285)-Net in Base 4 — Upper bound on s

There is no (111, 248, 1286)-net in base 4, because

1 times m-reduction [i] would yield (111, 247, 1286)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 52636 546968 970427 385376 051496 579213 611543 725704 550835 641947 846605 133314 449118 974860 686239 406585 437998 476593 757134 525803 204173 357627 445649 034788 194620 > 4²⁴⁷ [i]