MinT - Best Known (108−29, 108, s)-Nets in Base 5

Best Known (108−29, 108, s)-Nets in Base 5

Digital (79, 108, 400)-net over F₅, using

trace code for nets [i] based on digital (25, 54, 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 (79, 108, 1417)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁸, 1417, F₅, 29) (dual of [1417, 1309, 30]-code), using
- 1308 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (7, 3, 2, 2, 1, 1, 1, 1, 0, 1, 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, 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, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 25 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 35 times 0, 1, 38 times 0, 1, 39 times 0, 1, 42 times 0, 1, 45 times 0, 1, 48 times 0, 1, 50 times 0, 1, 53 times 0, 1, 57 times 0, 1, 60 times 0, 1, 64 times 0, 1, 68 times 0, 1, 72 times 0, 1, 77 times 0) [i] based on linear OA(5²⁹, 30, F₅, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,5)), using
  - dual of repetition code with length 30 [i]

There is no (79, 108, 332287)-net in base 5, because

1 times m-reduction [i] would yield (79, 107, 332287)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 616 313116 961661 952193 655593 835339 655675 786519 186032 114629 111010 746399 866825 > 5¹⁰⁷ [i]