MinT - Best Known (172−158, 172, s)-Nets in Base 4

Best Known (172−158, 172, s)-Nets in Base 4

(172−158, 172, 30)-Net over F₄ — Constructive and digital

Digital (14, 172, 30)-net over F₄, using

t-expansion [i] based on digital (13, 172, 30)-net over F₄, using
- net from sequence [i] based on digital (13, 29)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
    - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(172−158, 172, 33)-Net over F₄ — Digital

Digital (14, 172, 33)-net over F₄, using

t-expansion [i] based on digital (13, 172, 33)-net over F₄, using
- net from sequence [i] based on digital (13, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 33, using
    - Drinfeld modules of rank 1 [i]

(172−158, 172, 53)-Net in Base 4 — Upper bound on s

There is no (14, 172, 54)-net in base 4, because

14 times m-reduction [i] would yield (14, 158, 54)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁵⁸, 54, S₄, 3, 144), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 24 563850 913090 465947 542749 620949 849317 931102 007101 506854 390978 489955 526846 477366 325264 063999 770624 / 145 > 4¹⁵⁸ [i]