MinT - Best Known (58, 206, s)-Nets in Base 4

Best Known (58, 206, s)-Nets in Base 4

(58, 206, 66)-Net over F₄ — Constructive and digital

Digital (58, 206, 66)-net over F₄, using

t-expansion [i] based on digital (49, 206, 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]

(58, 206, 91)-Net over F₄ — Digital

Digital (58, 206, 91)-net over F₄, using

t-expansion [i] based on digital (50, 206, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(58, 206, 341)-Net over F₄ — Upper bound on s (digital)

There is no digital (58, 206, 342)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁶, 342, F₄, 148) (dual of [342, 136, 149]-code), but
- residual code [i] would yield OA(4⁵⁸, 193, S₄, 37), but
  - the linear programming bound shows that MÂ â‰¥ 161285 735386 400830 134445 178003 287186 762218 409717 533224 067961 483465 588736 000000 / 1 868030 896794 654733 447451 649501 557791 288931 > 4⁵⁸ [i]

(58, 206, 390)-Net in Base 4 — Upper bound on s

There is no (58, 206, 391)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11688 501628 880891 667541 065922 610867 657867 682745 844555 814986 259102 678368 267119 386468 046444 420389 579239 071062 343421 777588 467504 > 4²⁰⁶ [i]