MinT - Best Known (33, 112, s)-Nets in Base 4

Best Known (33, 112, s)-Nets in Base 4

(33, 112, 56)-Net over F₄ — Constructive and digital

Digital (33, 112, 56)-net over F₄, using

net from sequence [i] based on digital (33, 55)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
  - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(33, 112, 65)-Net over F₄ — Digital

Digital (33, 112, 65)-net over F₄, using

net from sequence [i] based on digital (33, 64)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
  - Drinfeld modules of rank 1 [i]

(33, 112, 215)-Net over F₄ — Upper bound on s (digital)

There is no digital (33, 112, 216)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹¹², 216, F₄, 79) (dual of [216, 104, 80]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹¹¹, 143, S₄, 79), but
    - the linear programming bound shows that MÂ â‰¥ 237 737030 554406 831140 331461 504711 505012 046426 036658 538250 578745 045288 416863 000860 792142 692352 / 26 111742 312416 213213 385625 > 4¹¹¹ [i]
  2. OA(4¹⁰⁴, 216, S₄, 73), but
    - discarding factors would yield OA(4¹⁰⁴, 150, S₄, 73), but
      - the linear programming bound shows that MÂ â‰¥ 32 028783 264027 932827 838658 827950 132017 443030 031565 779091 184319 519613 927802 631343 679332 066747 830675 439616 / 70932 376441 373736 310653 589982 582096 950125 > 4¹⁰⁴ [i]

(33, 112, 234)-Net in Base 4 — Upper bound on s

There is no (33, 112, 235)-net in base 4, because

1 times m-reduction [i] would yield (33, 111, 235)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 974965 211470 815411 274289 810557 902542 856926 295099 006724 017481 041760 > 4¹¹¹ [i]