MinT - Best Known (66, 66+194, s)-Nets in Base 4

Best Known (66, 66+194, s)-Nets in Base 4

(66, 66+194, 66)-Net over F₄ — Constructive and digital

Digital (66, 260, 66)-net over F₄, using

t-expansion [i] based on digital (49, 260, 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]

(66, 66+194, 99)-Net over F₄ — Digital

Digital (66, 260, 99)-net over F₄, using

t-expansion [i] based on digital (61, 260, 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]

(66, 66+194, 282)-Net over F₄ — Upper bound on s (digital)

There is no digital (66, 260, 283)-net over F₄, because

2 times m-reduction [i] would yield digital (66, 258, 283)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁸, 283, F₄, 192) (dual of [283, 25, 193]-code), but
  - residual code [i] would yield OA(4⁶⁶, 90, S₄, 48), but
    - the linear programming bound shows that MÂ â‰¥ 15992 796442 957684 354900 601418 173389 061260 037338 242442 330112 / 2 261259 518366 924075 > 4⁶⁶ [i]

(66, 66+194, 429)-Net in Base 4 — Upper bound on s

There is no (66, 260, 430)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 861527 907540 092752 680641 522854 698875 629895 520364 383909 583062 091382 166541 486233 316771 190527 246657 635701 583325 316062 674246 527418 783419 301268 975323 149320 445290 > 4²⁶⁰ [i]