MinT - Best Known (245−182, 245, s)-Nets in Base 4

Best Known (245−182, 245, s)-Nets in Base 4

(245−182, 245, 66)-Net over F₄ — Constructive and digital

Digital (63, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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]

(245−182, 245, 99)-Net over F₄ — Digital

Digital (63, 245, 99)-net over F₄, using

t-expansion [i] based on digital (61, 245, 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]

(245−182, 245, 283)-Net over F₄ — Upper bound on s (digital)

There is no digital (63, 245, 284)-net over F₄, because

2 times m-reduction [i] would yield digital (63, 243, 284)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴³, 284, F₄, 180) (dual of [284, 41, 181]-code), but
  - residual code [i] would yield OA(4⁶³, 103, S₄, 45), but
    - the linear programming bound shows that MÂ â‰¥ 30 963907 201042 572946 047598 093595 047914 220745 773823 416057 277667 026891 964416 / 314660 526989 952486 919791 531363 400925 > 4⁶³ [i]

(245−182, 245, 411)-Net in Base 4 — Upper bound on s

There is no (63, 245, 412)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3480 640611 141755 728804 216960 488036 037929 598342 310989 211746 669263 331562 874553 996522 869368 752983 434762 455007 956382 670831 011119 359094 080876 388420 994230 > 4²⁴⁵ [i]