MinT - Best Known (245−184, 245, s)-Nets in Base 4

Best Known (245−184, 245, s)-Nets in Base 4

(245−184, 245, 66)-Net over F₄ — Constructive and digital

Digital (61, 245, 66)-net over F₄, using

t-expansion [i] based on digital (49, 245, 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]

(245−184, 245, 99)-Net over F₄ — Digital

Digital (61, 245, 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]

(245−184, 245, 254)-Net over F₄ — Upper bound on s (digital)

There is no digital (61, 245, 255)-net over F₄, because

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]

(245−184, 245, 396)-Net in Base 4 — Upper bound on s

There is no (61, 245, 397)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3406 748771 314187 413647 895205 632523 335585 361553 363109 759964 027637 077018 575145 562842 762876 109394 616898 825672 206024 023266 133100 645712 000671 085508 281640 > 4²⁴⁵ [i]