MinT - Best Known (55, 55+140, s)-Nets in Base 4

Best Known (55, 55+140, s)-Nets in Base 4

(55, 55+140, 66)-Net over F₄ — Constructive and digital

Digital (55, 195, 66)-net over F₄, using

t-expansion [i] based on digital (49, 195, 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]

(55, 55+140, 91)-Net over F₄ — Digital

Digital (55, 195, 91)-net over F₄, using

t-expansion [i] based on digital (50, 195, 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]

(55, 55+140, 326)-Net over F₄ — Upper bound on s (digital)

There is no digital (55, 195, 327)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁹⁵, 327, F₄, 140) (dual of [327, 132, 141]-code), but
- residual code [i] would yield OA(4⁵⁵, 186, S₄, 35), but
  - the linear programming bound shows that MÂ â‰¥ 2227 845699 810146 058122 102603 602007 534656 013828 852609 458125 647380 480000 / 1 700436 329479 757404 993871 394563 863537 > 4⁵⁵ [i]

(55, 55+140, 371)-Net in Base 4 — Upper bound on s

There is no (55, 195, 372)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2836 660163 087054 166280 199138 841715 172893 659658 275794 637744 030408 812715 396739 405665 370499 840663 555786 394547 007447 672716 > 4¹⁹⁵ [i]