MinT - Best Known (37, 131, s)-Nets in Base 4

Best Known (37, 131, s)-Nets in Base 4

(37, 131, 56)-Net over F₄ — Constructive and digital

Digital (37, 131, 56)-net over F₄, using

t-expansion [i] based on digital (33, 131, 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]

(37, 131, 66)-Net over F₄ — Digital

Digital (37, 131, 66)-net over F₄, using

net from sequence [i] based on digital (37, 65)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 37 and N(F) ≥ 66, using
  - Drinfeld modules of rank 1 [i]

(37, 131, 185)-Net over F₄ — Upper bound on s (digital)

There is no digital (37, 131, 186)-net over F₄, because

1 times m-reduction [i] would yield digital (37, 130, 186)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹³⁰, 186, F₄, 93) (dual of [186, 56, 94]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(4¹²⁹, 150, S₄, 93), but
      - the linear programming bound shows that MÂ â‰¥ 334 695053 805711 397561 375492 015414 168229 977994 664798 746384 442171 245301 199155 917928 366979 678208 / 643 455449 300513 > 4¹²⁹ [i]
    2. OA(4⁵⁶, 186, S₄, 36), but
      - discarding factors would yield OA(4⁵⁶, 179, S₄, 36), but
        the linear programming bound shows that MÂ â‰¥ 31 511442 125419 258293 449040 901884 044621 696169 017943 480590 506370 111897 600000 / 5839 126320 766662 475522 447173 784673 305301 > 4⁵⁶ [i]

(37, 131, 254)-Net in Base 4 — Upper bound on s

There is no (37, 131, 255)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 467526 490573 582002 419198 225834 227543 603862 165358 020472 224783 082171 314770 457364 > 4¹³¹ [i]