MinT - Best Known (40, 40+104, s)-Nets in Base 4

Best Known (40, 40+104, s)-Nets in Base 4

(40, 40+104, 56)-Net over F₄ — Constructive and digital

Digital (40, 144, 56)-net over F₄, using

t-expansion [i] based on digital (33, 144, 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]

(40, 40+104, 75)-Net over F₄ — Digital

Digital (40, 144, 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]

(40, 40+104, 241)-Net over F₄ — Upper bound on s (digital)

There is no digital (40, 144, 242)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁴⁴, 242, F₄, 104) (dual of [242, 98, 105]-code), but
- residual code [i] would yield OA(4⁴⁰, 137, S₄, 26), but
  - the linear programming bound shows that MÂ â‰¥ 482 543072 349145 294988 866520 488882 831179 139598 254080 / 392 638906 322982 186151 177303 > 4⁴⁰ [i]

(40, 40+104, 272)-Net in Base 4 — Upper bound on s

There is no (40, 144, 273)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 531 536107 493834 843128 404534 221041 722655 313501 301484 205689 197262 011828 318066 304465 722688 > 4¹⁴⁴ [i]