MinT - Best Known (41, 41+111, s)-Nets in Base 4

Best Known (41, 41+111, s)-Nets in Base 4

(41, 41+111, 56)-Net over F₄ — Constructive and digital

Digital (41, 152, 56)-net over F₄, using

t-expansion [i] based on digital (33, 152, 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, 41+111, 75)-Net over F₄ — Digital

Digital (41, 152, 75)-net over F₄, using

t-expansion [i] based on digital (40, 152, 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, 41+111, 238)-Net over F₄ — Upper bound on s (digital)

There is no digital (41, 152, 239)-net over F₄, because

3 times m-reduction [i] would yield digital (41, 149, 239)-net over F₄, but
- 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, 41+111, 276)-Net in Base 4 — Upper bound on s

There is no (41, 152, 277)-net in base 4, because

1 times m-reduction [i] would yield (41, 151, 277)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 8 405889 471228 527631 652758 029025 711478 475808 942316 992996 751710 469289 371962 013697 757752 945920 > 4¹⁵¹ [i]