MinT - Best Known (56, 200, s)-Nets in Base 4

Best Known (56, 200, s)-Nets in Base 4

(56, 200, 66)-Net over F₄ — Constructive and digital

Digital (56, 200, 66)-net over F₄, using

t-expansion [i] based on digital (49, 200, 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]

(56, 200, 91)-Net over F₄ — Digital

Digital (56, 200, 91)-net over F₄, using

t-expansion [i] based on digital (50, 200, 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]

(56, 200, 323)-Net over F₄ — Upper bound on s (digital)

There is no digital (56, 200, 324)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁰⁰, 324, F₄, 144) (dual of [324, 124, 145]-code), but
- residual code [i] would yield OA(4⁵⁶, 179, S₄, 36), but
  - the linear programming bound shows that MÂ â‰¥ 31 511442 125419 258293 449040 901884 044621 696169 017943 480590 506370 111897 600000 / 5839 126320 766662 475522 447173 784673 305301 > 4⁵⁶ [i]

(56, 200, 376)-Net in Base 4 — Upper bound on s

There is no (56, 200, 377)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 725410 602329 665490 989920 527618 194510 411684 111119 570660 177195 278750 848573 660724 666489 300011 231778 970354 319556 188231 850405 > 4²⁰⁰ [i]