MinT - Best Known (65, 234, s)-Nets in Base 4

Best Known (65, 234, s)-Nets in Base 4

Digital (65, 234, 66)-net over F₄, using

t-expansion [i] based on digital (49, 234, 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]

Digital (65, 234, 99)-net over F₄, using

t-expansion [i] based on digital (61, 234, 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]

There is no digital (65, 234, 365)-net over F₄, because

1 times m-reduction [i] would yield digital (65, 233, 365)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³³, 365, F₄, 168) (dual of [365, 132, 169]-code), but
  - residual code [i] would yield OA(4⁶⁵, 196, S₄, 42), but
    - the linear programming bound shows that MÂ â‰¥ 396900 403316 059479 106808 484395 402977 413951 648726 549399 691598 760260 216224 172313 340280 832000 / 276 682488 299206 750393 648129 881706 051242 679804 397123 > 4⁶⁵ [i]

There is no (65, 234, 435)-net in base 4, because

1 times m-reduction [i] would yield (65, 233, 435)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 223 481453 264492 673832 858329 257452 238926 943425 053001 244303 298772 092042 077039 027872 806750 939476 721834 396510 193438 709698 630051 353228 788843 077360 > 4²³³ [i]