MinT - Best Known (125−89, 125, s)-Nets in Base 4

Best Known (125−89, 125, s)-Nets in Base 4

(125−89, 125, 56)-Net over F₄ — Constructive and digital

Digital (36, 125, 56)-net over F₄, using

t-expansion [i] based on digital (33, 125, 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]

(125−89, 125, 65)-Net over F₄ — Digital

Digital (36, 125, 65)-net over F₄, using

t-expansion [i] based on digital (33, 125, 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]

(125−89, 125, 190)-Net over F₄ — Upper bound on s (digital)

There is no digital (36, 125, 191)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹²⁵, 191, F₄, 89) (dual of [191, 66, 90]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(4¹²⁴, 148, S₄, 89), but
    - the linear programming bound shows that MÂ â‰¥ 8665 917257 296778 952342 423789 447146 428779 676886 749927 658316 397291 649757 237344 216595 862069 968896 / 15 234037 624603 222425 > 4¹²⁴ [i]
  2. OA(4⁶⁶, 191, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 28 766535 420054 304848 397361 421684 986456 320614 942528 963784 752261 977481 718335 232020 438414 046131 126272 000000 / 5209 403279 129645 739501 754543 221181 850628 166933 877437 442223 389477 > 4⁶⁶ [i]

(125−89, 125, 251)-Net in Base 4 — Upper bound on s

There is no (36, 125, 252)-net in base 4, because

1 times m-reduction [i] would yield (36, 124, 252)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 482 914408 740394 517158 583005 380335 490356 741743 841311 794512 345787 846610 298884 > 4¹²⁴ [i]