MinT - Best Known (106−28, 106, s)-Nets in Base 5

Best Known (106−28, 106, s)-Nets in Base 5

Digital (78, 106, 400)-net over F₅, using

trace code for nets [i] based on digital (25, 53, 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 (78, 106, 1529)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁶, 1529, F₅, 28) (dual of [1529, 1423, 29]-code), using
- 1422 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (7, 3, 2, 1, 1, 1, 1, 1, 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, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 4 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, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 34 times 0, 1, 36 times 0, 1, 38 times 0, 1, 41 times 0, 1, 44 times 0, 1, 46 times 0, 1, 49 times 0, 1, 53 times 0, 1, 56 times 0, 1, 59 times 0, 1, 63 times 0, 1, 67 times 0, 1, 71 times 0, 1, 76 times 0, 1, 81 times 0, 1, 86 times 0) [i] based on linear OA(5²⁸, 29, F₅, 28) (dual of [29, 1, 29]-code or 29-arc in PG(27,5)), using
  - dual of repetition code with length 29 [i]

There is no (78, 106, 296200)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 123 261528 810187 665424 129483 033239 374422 220158 794630 852131 544127 772873 613441 > 5¹⁰⁶ [i]