MinT - Best Known (64, 64+168, s)-Nets in Base 4

Best Known (64, 64+168, s)-Nets in Base 4

(64, 64+168, 66)-Net over F₄ — Constructive and digital

Digital (64, 232, 66)-net over F₄, using

t-expansion [i] based on digital (49, 232, 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]

(64, 64+168, 99)-Net over F₄ — Digital

Digital (64, 232, 99)-net over F₄, using

t-expansion [i] based on digital (61, 232, 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]

(64, 64+168, 348)-Net over F₄ — Upper bound on s (digital)

There is no digital (64, 232, 349)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³², 349, F₄, 168) (dual of [349, 117, 169]-code), but
- residual code [i] would yield OA(4⁶⁴, 180, S₄, 42), but
  - the linear programming bound shows that MÂ â‰¥ 387317 878288 940952 121032 892106 078533 601878 434569 146042 672353 835001 098548 766344 071344 765591 401529 344000 / 1135 320053 976181 914391 551986 355178 156290 618852 310180 915372 701779 > 4⁶⁴ [i]

(64, 64+168, 425)-Net in Base 4 — Upper bound on s

There is no (64, 232, 426)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48 267674 940246 676398 589618 071406 220844 214630 233831 149971 421066 573963 068932 128938 860996 060977 524577 226208 183136 340936 648061 232075 011464 640360 > 4²³² [i]