MinT - Best Known (104−27, 104, s)-Nets in Base 5

Best Known (104−27, 104, s)-Nets in Base 5

Digital (77, 104, 400)-net over F₅, using

trace code for nets [i] based on digital (25, 52, 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 (77, 104, 1662)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁴, 1662, F₅, 27) (dual of [1662, 1558, 28]-code), using
- 1557 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 3, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 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, 6 times 0, 1, 5 times 0, 1, 7 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, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 31 times 0, 1, 33 times 0, 1, 34 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 45 times 0, 1, 48 times 0, 1, 52 times 0, 1, 54 times 0, 1, 59 times 0, 1, 62 times 0, 1, 66 times 0, 1, 71 times 0, 1, 75 times 0, 1, 80 times 0, 1, 86 times 0, 1, 91 times 0, 1, 97 times 0) [i] based on linear OA(5²⁷, 28, F₅, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,5)), using
  - dual of repetition code with length 28 [i]

There is no (77, 104, 489026)-net in base 5, because

1 times m-reduction [i] would yield (77, 103, 489026)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 986101 702535 633097 889594 231410 700451 742501 511574 288996 891740 499880 258825 > 5¹⁰³ [i]