MinT - Best Known (62, 242, s)-Nets in Base 4

Best Known (62, 242, s)-Nets in Base 4

(62, 242, 66)-Net over F₄ — Constructive and digital

Digital (62, 242, 66)-net over F₄, using

t-expansion [i] based on digital (49, 242, 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]

(62, 242, 99)-Net over F₄ — Digital

Digital (62, 242, 99)-net over F₄, using

t-expansion [i] based on digital (61, 242, 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]

(62, 242, 268)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 242, 269)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁴², 269, F₄, 180) (dual of [269, 27, 181]-code), but
- residual code [i] would yield OA(4⁶², 88, S₄, 45), but
  - the linear programming bound shows that MÂ â‰¥ 1 115878 796922 411753 730265 232586 868317 882698 851925 622784 / 45512 145087 655961 > 4⁶² [i]

(62, 242, 404)-Net in Base 4 — Upper bound on s

There is no (62, 242, 405)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 50 435730 981595 432174 596519 128158 747186 904963 238226 660973 512465 247338 592450 967736 929823 355543 620356 604538 092650 640770 838370 927117 373210 862410 587872 > 4²⁴² [i]