MinT - Best Known (103, 103+29, s)-Nets in Base 3

Best Known (103, 103+29, s)-Nets in Base 3

Digital (103, 132, 600)-net over F₃, using

trace code for nets [i] based on digital (4, 33, 150)-net over F₈₁, using
- net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
    - a function field by SÃ©mirat [i]

Digital (103, 132, 1017)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(3¹³², 1017, F₃, 29) (dual of [1017, 885, 30]-code), using
- 884 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (10, 5, 3, 3, 2, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 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, 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, 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, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 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, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 32 times 0, 1, 34 times 0, 1, 35 times 0, 1, 37 times 0) [i] based on linear OA(3²⁹, 30, F₃, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
  - dual of repetition code with length 30 [i]

There is no (103, 132, 88076)-net in base 3, because

1 times m-reduction [i] would yield (103, 131, 88076)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 318 382769 533344 878696 405697 171087 658631 021088 560529 918596 280969 > 3¹³¹ [i]