MinT - Best Known (152−33, 152, s)-Nets in Base 3

Best Known (152−33, 152, s)-Nets in Base 3

Digital (119, 152, 640)-net over F₃, using

trace code for nets [i] based on digital (5, 38, 160)-net over F₈₁, using
- net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
    - a function field by SÃ©mirat [i]

Digital (119, 152, 1197)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹⁵², 1197, F₃, 33) (dual of [1197, 1045, 34]-code), using
- 1044 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (11, 6, 4, 2, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 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, 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, 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, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 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, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 35 times 0, 1, 37 times 0, 1, 38 times 0) [i] based on linear OA(3³³, 34, F₃, 33) (dual of [34, 1, 34]-code or 34-arc in PG(32,3)), using
  - dual of repetition code with length 34 [i]

There is no (119, 152, 108213)-net in base 3, because

1 times m-reduction [i] would yield (119, 151, 108213)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 110009 017296 671973 375160 342753 999581 944816 670824 778846 942017 216217 075137 > 3¹⁵¹ [i]