MinT - Best Known (82, 82+36, s)-Nets in Base 5

Best Known (82, 82+36, s)-Nets in Base 5

Digital (82, 118, 296)-net over F₅, using

8 times m-reduction [i] based on digital (82, 126, 296)-net over F₅, using
- trace code for nets [i] based on digital (19, 63, 148)-net over F₂₅, using
  - net from sequence [i] based on digital (19, 147)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 19 and N(F) ≥ 148, using
      - a function field by GarcÃa and Quoos [i]

Digital (82, 118, 808)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹¹⁸, 808, F₅, 36) (dual of [808, 690, 37]-code), using
- 689 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (8, 4, 2, 2, 2, 1, 1, 1, 1, 1, 0, 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, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 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, 7 times 0, 1, 8 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, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 19 times 0, 1, 19 times 0, 1, 20 times 0, 1, 22 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 34 times 0) [i] based on linear OA(5³⁶, 37, F₅, 36) (dual of [37, 1, 37]-code or 37-arc in PG(35,5)), using
  - dual of repetition code with length 37 [i]

There is no (82, 118, 72133)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 30099 505851 151493 014881 100842 333997 995551 523768 576712 550353 807050 942765 118178 833625 > 5¹¹⁸ [i]