MinT - Best Known (85, 120, s)-Nets in Base 5

Best Known (85, 120, s)-Nets in Base 5

Digital (85, 120, 400)-net over F₅, using

trace code for nets [i] based on digital (25, 60, 200)-net over F₂₅, using
- net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
    - a function field by GarcÃa and Quoos [i]

Digital (85, 120, 1009)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²⁰, 1009, F₅, 35) (dual of [1009, 889, 36]-code), using
- 888 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 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, 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, 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, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 12 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, 20 times 0, 1, 21 times 0, 1, 21 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 32 times 0, 1, 35 times 0, 1, 36 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0) [i] based on linear OA(5³⁵, 36, F₅, 35) (dual of [36, 1, 36]-code or 36-arc in PG(34,5)), using
  - dual of repetition code with length 36 [i]

There is no (85, 120, 140165)-net in base 5, because

1 times m-reduction [i] would yield (85, 119, 140165)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 150476 757508 131446 651338 390415 797443 918990 102174 152141 327033 322522 330542 359467 799765 > 5¹¹⁹ [i]