MinT - Best Known (65, 65+194, s)-Nets in Base 4

Best Known (65, 65+194, s)-Nets in Base 4

(65, 65+194, 66)-Net over F₄ — Constructive and digital

Digital (65, 259, 66)-net over F₄, using

t-expansion [i] based on digital (49, 259, 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+194, 99)-Net over F₄ — Digital

Digital (65, 259, 99)-net over F₄, using

t-expansion [i] based on digital (61, 259, 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+194, 272)-Net over F₄ — Upper bound on s (digital)

There is no digital (65, 259, 273)-net over F₄, because

2 times m-reduction [i] would yield digital (65, 257, 273)-net over F₄, but
- 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+194, 422)-Net in Base 4 — Upper bound on s

There is no (65, 259, 423)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 987965 758613 175833 028551 587744 693893 932031 029151 383527 926035 662697 541034 526957 445705 383451 259824 353331 254414 430564 790092 936819 019831 327288 088689 266432 903364 > 4²⁵⁹ [i]