MinT - Best Known (35, 35+86, s)-Nets in Base 4

Best Known (35, 35+86, s)-Nets in Base 4

(35, 35+86, 56)-Net over F₄ — Constructive and digital

Digital (35, 121, 56)-net over F₄, using

t-expansion [i] based on digital (33, 121, 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]

(35, 35+86, 65)-Net over F₄ — Digital

Digital (35, 121, 65)-net over F₄, using

t-expansion [i] based on digital (33, 121, 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]

(35, 35+86, 202)-Net over F₄ — Upper bound on s (digital)

There is no digital (35, 121, 203)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹²¹, 203, F₄, 86) (dual of [203, 82, 87]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹²⁰, 146, S₄, 86), but
    - the linear programming bound shows that MÂ â‰¥ 1676 785629 901870 831344 687640 931028 998195 565713 094557 086708 745951 276885 969823 837567 017285 582848 / 833 733992 529928 226625 > 4¹²⁰ [i]
  2. OA(4⁸², 203, S₄, 57), but
    - discarding factors would yield OA(4⁸², 147, S₄, 57), but
      - the linear programming bound shows that MÂ â‰¥ 317 037594 874104 788097 363934 385470 033055 949567 295348 251071 808033 395027 252472 917734 505583 665296 587433 584079 322091 728288 179828 060163 502846 553034 526452 257111 175653 922741 542240 051385 314719 039488 / 12 974914 487478 534018 893942 165576 018487 860552 476600 082829 690910 525536 318801 189236 245587 024898 169738 332511 320472 102082 003724 381476 705681 480945 > 4⁸² [i]

(35, 35+86, 244)-Net in Base 4 — Upper bound on s

There is no (35, 121, 245)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 541460 917145 996894 414259 569307 018077 224813 619796 000852 360038 123435 454552 > 4¹²¹ [i]