MinT - Best Known (42, 42+116, s)-Nets in Base 4

Best Known (42, 42+116, s)-Nets in Base 4

(42, 42+116, 56)-Net over F₄ — Constructive and digital

Digital (42, 158, 56)-net over F₄, using

t-expansion [i] based on digital (33, 158, 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]

(42, 42+116, 75)-Net over F₄ — Digital

Digital (42, 158, 75)-net over F₄, using

t-expansion [i] based on digital (40, 158, 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]

(42, 42+116, 222)-Net over F₄ — Upper bound on s (digital)

There is no digital (42, 158, 223)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁸, 223, F₄, 116) (dual of [223, 65, 117]-code), but
- residual code [i] would yield OA(4⁴², 106, S₄, 29), but
  - the linear programming bound shows that MÂ â‰¥ 11 515430 185021 615009 307015 123093 343331 130678 608727 482845 042118 230016 / 573326 282922 594402 886717 422034 435890 047071 > 4⁴² [i]

(42, 42+116, 281)-Net in Base 4 — Upper bound on s

There is no (42, 158, 282)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 148750 864232 869938 420539 847404 001449 439175 121663 653487 701340 662752 423207 838224 438060 788329 789120 > 4¹⁵⁸ [i]