MinT - Best Known (224−160, 224, s)-Nets in Base 4

Best Known (224−160, 224, s)-Nets in Base 4

(224−160, 224, 66)-Net over F₄ — Constructive and digital

Digital (64, 224, 66)-net over F₄, using

t-expansion [i] based on digital (49, 224, 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]

(224−160, 224, 99)-Net over F₄ — Digital

Digital (64, 224, 99)-net over F₄, using

t-expansion [i] based on digital (61, 224, 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]

(224−160, 224, 384)-Net over F₄ — Upper bound on s (digital)

There is no digital (64, 224, 385)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²⁴, 385, F₄, 160) (dual of [385, 161, 161]-code), but
- residual code [i] would yield OA(4⁶⁴, 224, S₄, 40), but
  - the linear programming bound shows that MÂ â‰¥ 13361 740924 714558 183981 708538 240631 845789 107040 889033 084067 601409 088040 474771 456000 / 36 499780 543012 793452 506240 719222 345774 789003 > 4⁶⁴ [i]

(224−160, 224, 431)-Net in Base 4 — Upper bound on s

There is no (64, 224, 432)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 765 905534 040528 923804 046003 102875 479818 603149 565003 189800 421836 143300 137263 932007 193111 140599 939822 821657 019998 487553 550332 697116 645860 > 4²²⁴ [i]