MinT - Best Known (257−222, 257, s)-Nets in Base 4

Best Known (257−222, 257, s)-Nets in Base 4

(257−222, 257, 56)-Net over F₄ — Constructive and digital

Digital (35, 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]

(257−222, 257, 65)-Net over F₄ — Digital

Digital (35, 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]

(257−222, 257, 124)-Net in Base 4 — Upper bound on s

There is no (35, 257, 125)-net in base 4, because

11 times m-reduction [i] would yield (35, 246, 125)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²⁴⁶, 125, S₄, 2, 211), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 869494 380222 412684 222253 513219 831024 807545 543480 189084 706499 269650 535539 669991 678640 868618 274068 291743 204275 471383 143861 823333 770000 807529 790390 140928 / 53 > 4²⁴⁶ [i]