MinT - Best Known (247−135, 247, s)-Nets in Base 4

Best Known (247−135, 247, s)-Nets in Base 4

(247−135, 247, 130)-Net over F₄ — Constructive and digital

Digital (112, 247, 130)-net over F₄, using

t-expansion [i] based on digital (105, 247, 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]

(247−135, 247, 165)-Net over F₄ — Digital

Digital (112, 247, 165)-net over F₄, using

t-expansion [i] based on digital (109, 247, 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]

(247−135, 247, 1341)-Net in Base 4 — Upper bound on s

There is no (112, 247, 1342)-net in base 4, because

1 times m-reduction [i] would yield (112, 246, 1342)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13279 909887 492575 825927 679648 852064 352679 984615 460438 478028 141886 325371 167259 091460 530179 419024 950219 831176 207641 732495 065484 113496 135351 301199 662600 > 4²⁴⁶ [i]