MinT - Best Known (250−189, 250, s)-Nets in Base 4

Best Known (250−189, 250, s)-Nets in Base 4

(250−189, 250, 66)-Net over F₄ — Constructive and digital

Digital (61, 250, 66)-net over F₄, using

t-expansion [i] based on digital (49, 250, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(250−189, 250, 99)-Net over F₄ — Digital

Digital (61, 250, 99)-net over F₄, using

net from sequence [i] based on digital (61, 98)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
  - Drinfeld modules of rank 1 [i]

(250−189, 250, 254)-Net over F₄ — Upper bound on s (digital)

There is no digital (61, 250, 255)-net over F₄, because

5 times m-reduction [i] would yield digital (61, 245, 255)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁵, 255, F₄, 184) (dual of [255, 10, 185]-code), but
  - residual code [i] would yield linear OA(4⁶¹, 70, F₄, 46) (dual of [70, 9, 47]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(250−189, 250, 395)-Net in Base 4 — Upper bound on s

There is no (61, 250, 396)-net in base 4, because

1 times m-reduction [i] would yield (61, 249, 396)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 849733 602787 781297 888077 147381 150546 280293 790196 805395 064576 085435 101159 048419 190539 256196 873915 654363 012032 694766 237967 384137 654613 747871 790257 488864 > 4²⁴⁹ [i]