MinT - Best Known (57, 57+144, s)-Nets in Base 4

Best Known (57, 57+144, s)-Nets in Base 4

(57, 57+144, 66)-Net over F₄ — Constructive and digital

Digital (57, 201, 66)-net over F₄, using

t-expansion [i] based on digital (49, 201, 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]

(57, 57+144, 91)-Net over F₄ — Digital

Digital (57, 201, 91)-net over F₄, using

t-expansion [i] based on digital (50, 201, 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]

(57, 57+144, 339)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 201, 340)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁰¹, 340, F₄, 144) (dual of [340, 139, 145]-code), but
- residual code [i] would yield OA(4⁵⁷, 195, S₄, 36), but
  - the linear programming bound shows that MÂ â‰¥ 494 490067 436667 455875 598203 818411 591542 653847 684514 372082 636620 702806 966272 / 22309 972171 916724 959618 830756 395146 773299 > 4⁵⁷ [i]

(57, 57+144, 385)-Net in Base 4 — Upper bound on s

There is no (57, 201, 386)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12 072154 740893 860680 503668 635395 582364 270436 662646 636628 962624 963729 682252 092437 229984 920359 043426 134720 936900 287776 546736 > 4²⁰¹ [i]