MinT - Best Known (57, 209, s)-Nets in Base 4

Best Known (57, 209, s)-Nets in Base 4

(57, 209, 66)-Net over F₄ — Constructive and digital

Digital (57, 209, 66)-net over F₄, using

t-expansion [i] based on digital (49, 209, 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, 209, 91)-Net over F₄ — Digital

Digital (57, 209, 91)-net over F₄, using

t-expansion [i] based on digital (50, 209, 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, 209, 306)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 209, 307)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁹, 307, F₄, 152) (dual of [307, 98, 153]-code), but
- residual code [i] would yield OA(4⁵⁷, 154, S₄, 38), but
  - the linear programming bound shows that MÂ â‰¥ 797207 451366 565965 705395 219386 483301 387169 888973 699524 917197 426960 177348 462205 201075 404800 / 37 245253 436418 517434 670471 565022 824982 371808 117558 951421 > 4⁵⁷ [i]

(57, 209, 379)-Net in Base 4 — Upper bound on s

There is no (57, 209, 380)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 726670 787927 435726 748307 987581 672018 577175 014526 438986 055133 620334 780929 422290 454785 586232 164752 508902 588347 665831 638936 188644 > 4²⁰⁹ [i]