MinT - Best Known (41, 149, s)-Nets in Base 4

Best Known (41, 149, s)-Nets in Base 4

(41, 149, 56)-Net over F₄ — Constructive and digital

Digital (41, 149, 56)-net over F₄, using

t-expansion [i] based on digital (33, 149, 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]

(41, 149, 75)-Net over F₄ — Digital

Digital (41, 149, 75)-net over F₄, using

t-expansion [i] based on digital (40, 149, 75)-net over F₄, using
- net from sequence [i] based on digital (40, 74)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 40 and N(F) ≥ 75, using
    - Drinfeld modules of rank 1 [i]

(41, 149, 238)-Net over F₄ — Upper bound on s (digital)

There is no digital (41, 149, 239)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁹, 239, F₄, 108) (dual of [239, 90, 109]-code), but
- residual code [i] would yield OA(4⁴¹, 130, S₄, 27), but
  - the linear programming bound shows that MÂ â‰¥ 21 043123 812010 240591 636510 926917 514688 253132 800000 / 4 066822 030762 196031 534869 > 4⁴¹ [i]

(41, 149, 278)-Net in Base 4 — Upper bound on s

There is no (41, 149, 279)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 594849 563778 836817 998846 713907 841745 550409 767125 168169 489088 094244 987906 941229 828995 150600 > 4¹⁴⁹ [i]