MinT - Best Known (65, 65+156, s)-Nets in Base 4

Best Known (65, 65+156, s)-Nets in Base 4

(65, 65+156, 66)-Net over F₄ — Constructive and digital

Digital (65, 221, 66)-net over F₄, using

t-expansion [i] based on digital (49, 221, 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+156, 99)-Net over F₄ — Digital

Digital (65, 221, 99)-net over F₄, using

t-expansion [i] based on digital (61, 221, 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+156, 433)-Net over F₄ — Upper bound on s (digital)

There is no digital (65, 221, 434)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²¹, 434, F₄, 156) (dual of [434, 213, 157]-code), but
- residual code [i] would yield OA(4⁶⁵, 277, S₄, 39), but
  - the linear programming bound shows that MÂ â‰¥ 114860 870016 374579 218653 717961 476899 219304 784975 141224 743527 918706 993971 303087 603712 / 81 574206 852274 374488 234675 560995 174637 976327 > 4⁶⁵ [i]

(65, 65+156, 443)-Net in Base 4 — Upper bound on s

There is no (65, 221, 444)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11 360605 748267 414006 716214 265251 091416 172483 640073 054366 781144 035522 594163 661138 521814 110156 754464 077791 374857 785933 175230 920101 519320 > 4²²¹ [i]