MinT - Best Known (127−91, 127, s)-Nets in Base 4

Best Known (127−91, 127, s)-Nets in Base 4

(127−91, 127, 56)-Net over F₄ — Constructive and digital

Digital (36, 127, 56)-net over F₄, using

t-expansion [i] based on digital (33, 127, 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]

(127−91, 127, 65)-Net over F₄ — Digital

Digital (36, 127, 65)-net over F₄, using

t-expansion [i] based on digital (33, 127, 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]

(127−91, 127, 182)-Net over F₄ — Upper bound on s (digital)

There is no digital (36, 127, 183)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹²⁷, 183, F₄, 91) (dual of [183, 56, 92]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹²⁶, 147, S₄, 91), but
    - the linear programming bound shows that MÂ â‰¥ 340456 058530 442352 522099 250356 400004 547517 225644 691214 440472 273914 376101 952655 553716 027392 / 33 253308 357995 > 4¹²⁶ [i]
  2. OA(4⁵⁶, 183, 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]

(127−91, 127, 249)-Net in Base 4 — Upper bound on s

There is no (36, 127, 250)-net in base 4, because

1 times m-reduction [i] would yield (36, 126, 250)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7579 865651 780003 565952 436699 234036 068015 179699 405267 041938 622971 979202 386356 > 4¹²⁶ [i]