MinT - Best Known (124−32, 124, s)-Nets in Base 5

Best Known (124−32, 124, s)-Nets in Base 5

Digital (92, 124, 408)-net over F₅, using

trace code for nets [i] based on digital (30, 62, 204)-net over F₂₅, using
- net from sequence [i] based on digital (30, 203)-sequence over F₂₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 30 and N(F) ≥ 204, using
    - a function field by GarcÃa and Quoos [i]

Digital (92, 124, 1956)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²⁴, 1956, F₅, 32) (dual of [1956, 1832, 33]-code), using
- 1831 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (8, 3, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 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, 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, 6 times 0, 1, 6 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, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 24 times 0, 1, 26 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 32 times 0, 1, 35 times 0, 1, 37 times 0, 1, 38 times 0, 1, 41 times 0, 1, 43 times 0, 1, 45 times 0, 1, 48 times 0, 1, 51 times 0, 1, 53 times 0, 1, 57 times 0, 1, 59 times 0, 1, 63 times 0, 1, 66 times 0, 1, 70 times 0, 1, 74 times 0, 1, 78 times 0, 1, 82 times 0, 1, 86 times 0, 1, 91 times 0, 1, 97 times 0) [i] based on linear OA(5³², 33, F₅, 32) (dual of [33, 1, 33]-code or 33-arc in PG(31,5)), using
  - dual of repetition code with length 33 [i]

There is no (92, 124, 444105)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 470 208854 172764 706993 737410 818602 838522 298690 567784 098664 448614 259838 081853 508899 099585 > 5¹²⁴ [i]