MinT - Best Known (65, 65+192, s)-Nets in Base 4

Best Known (65, 65+192, s)-Nets in Base 4

(65, 65+192, 66)-Net over F₄ — Constructive and digital

Digital (65, 257, 66)-net over F₄, using

t-expansion [i] based on digital (49, 257, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(65, 65+192, 99)-Net over F₄ — Digital

Digital (65, 257, 99)-net over F₄, using

t-expansion [i] based on digital (61, 257, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(65, 65+192, 272)-Net over F₄ — Upper bound on s (digital)

There is no digital (65, 257, 273)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁷, 273, F₄, 192) (dual of [273, 16, 193]-code), but
- residual code [i] would yield OA(4⁶⁵, 80, S₄, 48), but
  - the linear programming bound shows that MÂ â‰¥ 11 748891 149276 075699 951965 511184 653803 268463 394816 / 8602 297165 > 4⁶⁵ [i]

(65, 65+192, 422)-Net in Base 4 — Upper bound on s

There is no (65, 257, 423)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 55438 923910 194935 279983 574927 122155 435881 064636 918611 939559 459894 405534 458876 854830 859722 856856 648763 867262 959694 161023 755891 328799 992931 183177 867895 082470 > 4²⁵⁷ [i]