MinT - Best Known (43, 43+118, s)-Nets in Base 4

Best Known (43, 43+118, s)-Nets in Base 4

(43, 43+118, 56)-Net over F₄ — Constructive and digital

Digital (43, 161, 56)-net over F₄, using

t-expansion [i] based on digital (33, 161, 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]

(43, 43+118, 75)-Net over F₄ — Digital

Digital (43, 161, 75)-net over F₄, using

t-expansion [i] based on digital (40, 161, 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]

(43, 43+118, 238)-Net over F₄ — Upper bound on s (digital)

There is no digital (43, 161, 239)-net over F₄, because

2 times m-reduction [i] would yield digital (43, 159, 239)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4¹⁵⁹, 239, F₄, 116) (dual of [239, 80, 117]-code), but
  - residual code [i] would yield OA(4⁴³, 122, S₄, 29), but
    - the linear programming bound shows that MÂ â‰¥ 14974 882928 795402 502879 363014 902325 642932 640714 588160 / 186 039615 985017 664535 163073 > 4⁴³ [i]

(43, 43+118, 288)-Net in Base 4 — Upper bound on s

There is no (43, 161, 289)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 943159 866210 574845 370519 924422 633822 169808 358025 772915 987183 058143 818586 728347 405301 588842 215680 > 4¹⁶¹ [i]