MinT - Best Known (122−87, 122, s)-Nets in Base 4

Best Known (122−87, 122, s)-Nets in Base 4

(122−87, 122, 56)-Net over F₄ — Constructive and digital

Digital (35, 122, 56)-net over F₄, using

t-expansion [i] based on digital (33, 122, 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]

(122−87, 122, 65)-Net over F₄ — Digital

Digital (35, 122, 65)-net over F₄, using

t-expansion [i] based on digital (33, 122, 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]

(122−87, 122, 188)-Net over F₄ — Upper bound on s (digital)

There is no digital (35, 122, 189)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹²², 189, F₄, 87) (dual of [189, 67, 88]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹²¹, 145, S₄, 87), but
    - the linear programming bound shows that MÂ â‰¥ 241502 590268 189986 517960 819451 991263 770225 532833 261695 725476 600361 862422 859217 680809 852928 / 21697 738822 577169 > 4¹²¹ [i]
  2. OA(4⁶⁷, 189, S₄, 44), but
    - discarding factors would yield OA(4⁶⁷, 185, S₄, 44), but
      - the linear programming bound shows that MÂ â‰¥ 27 922736 614479 963563 498030 039555 280180 497392 114764 937434 686895 163379 799248 329552 391127 539087 989670 936576 / 1212 198916 755108 948361 633672 026124 789528 916769 954260 490443 945723 > 4⁶⁷ [i]

(122−87, 122, 244)-Net in Base 4 — Upper bound on s

There is no (35, 122, 245)-net in base 4, because

1 times m-reduction [i] would yield (35, 121, 245)-net in base 4, but
- 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]