MinT - Best Known (108−31, 108, s)-Nets in Base 5

Best Known (108−31, 108, s)-Nets in Base 5

Digital (77, 108, 296)-net over F₅, using

8 times m-reduction [i] based on digital (77, 116, 296)-net over F₅, using
- trace code for nets [i] based on digital (19, 58, 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 (77, 108, 1004)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹⁰⁸, 1004, F₅, 31) (dual of [1004, 896, 32]-code), using
- 895 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (7, 4, 2, 1, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 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, 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, 7 times 0, 1, 6 times 0, 1, 8 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, 12 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 35 times 0, 1, 38 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0, 1, 47 times 0, 1, 50 times 0) [i] based on linear OA(5³¹, 32, F₅, 31) (dual of [32, 1, 32]-code or 32-arc in PG(30,5)), using
  - dual of repetition code with length 32 [i]

There is no (77, 108, 155476)-net in base 5, because

1 times m-reduction [i] would yield (77, 107, 155476)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 616 312586 691135 117110 637469 353517 863896 023583 594580 346116 424356 454465 797585 > 5¹⁰⁷ [i]