MinT - Best Known (221−160, 221, s)-Nets in Base 4

Best Known (221−160, 221, s)-Nets in Base 4

(221−160, 221, 66)-Net over F₄ — Constructive and digital

Digital (61, 221, 66)-net over F₄, using

t-expansion [i] based on digital (49, 221, 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]

(221−160, 221, 99)-Net over F₄ — Digital

Digital (61, 221, 99)-net over F₄, using

net from sequence [i] based on digital (61, 98)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
  - Drinfeld modules of rank 1 [i]

(221−160, 221, 335)-Net over F₄ — Upper bound on s (digital)

There is no digital (61, 221, 336)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²¹, 336, F₄, 160) (dual of [336, 115, 161]-code), but
- residual code [i] would yield OA(4⁶¹, 175, S₄, 40), but
  - the linear programming bound shows that MÂ â‰¥ 1 798315 977868 370852 713534 848248 812726 194497 404051 298068 679657 800761 225918 150102 236050 489344 / 313133 764813 349339 900431 732581 184165 925845 010666 409433 > 4⁶¹ [i]

(221−160, 221, 406)-Net in Base 4 — Upper bound on s

There is no (61, 221, 407)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 801880 919960 343477 642916 434495 625261 706408 718087 879623 892176 559850 055178 642960 646813 017386 624613 445349 184574 937427 070598 083527 915266 > 4²²¹ [i]