MinT - Best Known (87, 120, s)-Nets in Base 4

Best Known (87, 120, s)-Nets in Base 4

Digital (87, 120, 531)-net over F₄, using

trace code for nets [i] based on digital (7, 40, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

Digital (87, 120, 788)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²⁰, 788, F₄, 33) (dual of [788, 668, 34]-code), using
- 667 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (9, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 23 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0) [i] based on linear OA(4³³, 34, F₄, 33) (dual of [34, 1, 34]-code or 34-arc in PG(32,4)), using
  - dual of repetition code with length 34 [i]

There is no (87, 120, 68102)-net in base 4, because

1 times m-reduction [i] would yield (87, 119, 68102)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 441761 189649 187496 859962 020734 899886 992729 876716 951180 614813 272147 466208 > 4¹¹⁹ [i]