MinT - Best Known (61, 61+157, s)-Nets in Base 4

Best Known (61, 61+157, s)-Nets in Base 4

Digital (61, 218, 66)-net over F₄, using

t-expansion [i] based on digital (49, 218, 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 (61, 218, 99)-net over F₄, using

There is no digital (61, 218, 357)-net over F₄, because

1 times m-reduction [i] would yield digital (61, 217, 357)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²¹⁷, 357, F₄, 156) (dual of [357, 140, 157]-code), but
  - residual code [i] would yield OA(4⁶¹, 200, S₄, 39), but
    - the linear programming bound shows that MÂ â‰¥ 2 016494 570162 278735 518158 542717 648360 131511 752234 027270 947252 517739 366352 683008 / 364228 998441 829554 049476 381326 982618 927461 > 4⁶¹ [i]

There is no (61, 218, 410)-net in base 4, because

1 times m-reduction [i] would yield (61, 217, 410)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 48151 976854 521574 664503 682624 233388 017880 873361 564481 249212 808001 078779 687899 812153 675106 750392 974056 844636 363659 840966 807972 050608 > 4²¹⁷ [i]