MinT - Best Known (64, 64+187, s)-Nets in Base 4

Best Known (64, 64+187, s)-Nets in Base 4

(64, 64+187, 66)-Net over F₄ — Constructive and digital

Digital (64, 251, 66)-net over F₄, using

t-expansion [i] based on digital (49, 251, 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]

(64, 64+187, 99)-Net over F₄ — Digital

Digital (64, 251, 99)-net over F₄, using

t-expansion [i] based on digital (61, 251, 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]

(64, 64+187, 281)-Net over F₄ — Upper bound on s (digital)

There is no digital (64, 251, 282)-net over F₄, because

3 times m-reduction [i] would yield digital (64, 248, 282)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁸, 282, F₄, 184) (dual of [282, 34, 185]-code), but
  - residual code [i] would yield OA(4⁶⁴, 97, S₄, 46), but
    - the linear programming bound shows that MÂ â‰¥ 17 551672 443269 830978 915105 133499 110631 230918 120257 846151 906745 909248 / 44694 499640 915429 319268 220967 > 4⁶⁴ [i]

(64, 64+187, 417)-Net in Base 4 — Upper bound on s

There is no (64, 251, 418)-net in base 4, because

1 times m-reduction [i] would yield (64, 250, 418)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 614971 491804 404910 233279 292176 934991 634623 234937 005381 222472 548028 336872 718406 252443 452617 519550 270735 259084 463954 721937 121759 266049 839418 435590 142160 > 4²⁵⁰ [i]