MinT - Best Known (257−221, 257, s)-Nets in Base 4

Best Known (257−221, 257, s)-Nets in Base 4

(257−221, 257, 56)-Net over F₄ — Constructive and digital

Digital (36, 257, 56)-net over F₄, using

t-expansion [i] based on digital (33, 257, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(257−221, 257, 65)-Net over F₄ — Digital

Digital (36, 257, 65)-net over F₄, using

t-expansion [i] based on digital (33, 257, 65)-net over F₄, using
- net from sequence [i] based on digital (33, 64)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(257−221, 257, 127)-Net in Base 4 — Upper bound on s

There is no (36, 257, 128)-net in base 4, because

5 times m-reduction [i] would yield (36, 252, 128)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁵², 128, S₄, 2, 216), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 13407 807929 942597 099574 024998 205846 127479 365820 592393 377723 561443 721764 030073 546976 801874 298166 903427 690031 858186 486050 853753 882811 946569 946433 649006 084096 / 217 > 4²⁵² [i]