MinT - Best Known (255−194, 255, s)-Nets in Base 4

Best Known (255−194, 255, s)-Nets in Base 4

(255−194, 255, 66)-Net over F₄ — Constructive and digital

Digital (61, 255, 66)-net over F₄, using

t-expansion [i] based on digital (49, 255, 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]

(255−194, 255, 99)-Net over F₄ — Digital

Digital (61, 255, 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]

(255−194, 255, 254)-Net over F₄ — Upper bound on s (digital)

There is no digital (61, 255, 255)-net over F₄, because

10 times m-reduction [i] would yield digital (61, 245, 255)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁵, 255, F₄, 184) (dual of [255, 10, 185]-code), but
  - residual code [i] would yield linear OA(4⁶¹, 70, F₄, 46) (dual of [70, 9, 47]-code), but
    - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(255−194, 255, 394)-Net in Base 4 — Upper bound on s

There is no (61, 255, 395)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3453 892113 974933 502210 709264 259782 216574 237025 460115 961961 940842 674961 782965 416468 222450 610998 502276 714750 996595 819398 365464 948614 652883 114513 008070 060456 > 4²⁵⁵ [i]