MinT - Best Known (57, 57+156, s)-Nets in Base 4

Best Known (57, 57+156, s)-Nets in Base 4

(57, 57+156, 66)-Net over F₄ — Constructive and digital

Digital (57, 213, 66)-net over F₄, using

t-expansion [i] based on digital (49, 213, 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+156, 91)-Net over F₄ — Digital

Digital (57, 213, 91)-net over F₄, using

t-expansion [i] based on digital (50, 213, 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+156, 290)-Net over F₄ — Upper bound on s (digital)

There is no digital (57, 213, 291)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²¹³, 291, F₄, 156) (dual of [291, 78, 157]-code), but
- residual code [i] would yield OA(4⁵⁷, 134, S₄, 39), but
  - the linear programming bound shows that MÂ â‰¥ 2118 872152 947773 735205 614860 840104 993723 173522 547123 836257 668222 520389 144284 698869 497944 229365 678080 000000 / 96823 132929 101376 060431 157559 643789 049324 330608 249156 649224 680559 609039 > 4⁵⁷ [i]

(57, 57+156, 377)-Net in Base 4 — Upper bound on s

There is no (57, 213, 378)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 192 523968 013779 794296 351827 005194 500005 779641 483465 332272 102742 227672 262297 753380 288085 896081 730975 642495 413503 404649 154519 988440 > 4²¹³ [i]