MinT - Best Known (39, 39+19, s)-Nets in Base 5

Best Known (39, 39+19, s)-Nets in Base 5

Digital (39, 58, 252)-net over F₅, using

trace code for nets [i] based on digital (10, 29, 126)-net over F₂₅, using
- net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
    - the Hermitian function field over F₂₅ [i]

Digital (39, 58, 347)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5⁵⁸, 347, F₅, 19) (dual of [347, 289, 20]-code), using
- 288 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 3, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 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, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 22 times 0, 1, 25 times 0, 1, 27 times 0) [i] based on linear OA(5¹⁹, 20, F₅, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,5)), using
  - dual of repetition code with length 20 [i]

There is no (39, 58, 27696)-net in base 5, because

1 times m-reduction [i] would yield (39, 57, 27696)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 6940 842876 151001 787866 836201 008640 361665 > 5⁵⁷ [i]