MinT - Best Known (63, 63+187, s)-Nets in Base 4

Best Known (63, 63+187, s)-Nets in Base 4

(63, 63+187, 66)-Net over F₄ — Constructive and digital

Digital (63, 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]

(63, 63+187, 99)-Net over F₄ — Digital

Digital (63, 250, 99)-net over F₄, using

t-expansion [i] based on 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]

(63, 63+187, 268)-Net over F₄ — Upper bound on s (digital)

There is no digital (63, 250, 269)-net over F₄, because

3 times m-reduction [i] would yield digital (63, 247, 269)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁷, 269, F₄, 184) (dual of [269, 22, 185]-code), but
  - residual code [i] would yield OA(4⁶³, 84, S₄, 46), but
    - the linear programming bound shows that MÂ â‰¥ 894450 329286 322020 798598 021001 283152 584338 810439 991296 / 9348 861618 951283 > 4⁶³ [i]

(63, 63+187, 410)-Net in Base 4 — Upper bound on s

There is no (63, 250, 411)-net in base 4, because

1 times m-reduction [i] would yield (63, 249, 411)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 939758 150686 263126 379760 298007 376399 557078 050801 022620 277724 322280 574158 777007 108951 933626 359399 172755 423466 126981 378696 955060 239671 967066 933227 968320 > 4²⁴⁹ [i]