MinT - Best Known (62, 62+182, s)-Nets in Base 4

Best Known (62, 62+182, s)-Nets in Base 4

(62, 62+182, 66)-Net over F₄ — Constructive and digital

Digital (62, 244, 66)-net over F₄, using

t-expansion [i] based on digital (49, 244, 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, 62+182, 99)-Net over F₄ — Digital

Digital (62, 244, 99)-net over F₄, using

t-expansion [i] based on digital (61, 244, 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, 62+182, 268)-Net over F₄ — Upper bound on s (digital)

There is no digital (62, 244, 269)-net over F₄, because

2 times m-reduction [i] would yield digital (62, 242, 269)-net over F₄, but
- 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, 62+182, 404)-Net in Base 4 — Upper bound on s

There is no (62, 244, 405)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 912 423222 714578 481409 769990 408140 009382 078683 667809 558837 402495 639922 830756 140533 867732 244689 571811 748807 136653 221891 921476 009842 775062 374852 892992 > 4²⁴⁴ [i]