MinT - Best Known (73, 73+180, s)-Nets in Base 4

Best Known (73, 73+180, s)-Nets in Base 4

(73, 73+180, 104)-Net over F₄ — Constructive and digital

Digital (73, 253, 104)-net over F₄, using

net from sequence [i] based on digital (73, 103)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
  - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(73, 73+180, 112)-Net over F₄ — Digital

Digital (73, 253, 112)-net over F₄, using

net from sequence [i] based on digital (73, 111)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
  - Drinfeld modules of rank 1 [i]

(73, 73+180, 448)-Net over F₄ — Upper bound on s (digital)

There is no digital (73, 253, 449)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵³, 449, F₄, 180) (dual of [449, 196, 181]-code), but
- residual code [i] would yield OA(4⁷³, 268, S₄, 45), but
  - the linear programming bound shows that MÂ â‰¥ 472 872631 524693 616571 263304 632879 333287 600071 846650 492027 623804 614698 703694 083741 742382 499726 622720 / 5 268581 092830 135391 831262 242618 253418 847780 919324 019929 > 4⁷³ [i]

(73, 73+180, 492)-Net in Base 4 — Upper bound on s

There is no (73, 253, 493)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 241 714853 751193 003143 292367 525004 112521 107815 914337 823183 117456 529426 404993 059758 455494 441631 185334 222602 989594 765318 548996 554310 398791 240292 702083 046132 > 4²⁵³ [i]