MinT - Best Known (76, 76+32, s)-Nets in Base 7

Best Known (76, 76+32, s)-Nets in Base 7

Digital (76, 108, 688)-net over F₇, using

2 times m-reduction [i] based on digital (76, 110, 688)-net over F₇, using
- trace code for nets [i] based on digital (21, 55, 344)-net over F₄₉, using
  - net from sequence [i] based on digital (21, 343)-sequence over F₄₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄₉ with g(F) = 21 and N(F) ≥ 344, using
      - the Hermitian function field over F₄₉ [i]

Digital (76, 108, 1836)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7¹⁰⁸, 1836, F₇, 32) (dual of [1836, 1728, 33]-code), using
- 1727 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (6, 3, 1, 1, 1, 1, 1, 1, 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, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 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, 19 times 0, 1, 19 times 0, 1, 21 times 0, 1, 23 times 0, 1, 24 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 32 times 0, 1, 34 times 0, 1, 36 times 0, 1, 38 times 0, 1, 41 times 0, 1, 44 times 0, 1, 47 times 0, 1, 50 times 0, 1, 54 times 0, 1, 57 times 0, 1, 61 times 0, 1, 65 times 0, 1, 69 times 0, 1, 74 times 0, 1, 79 times 0, 1, 84 times 0, 1, 90 times 0, 1, 96 times 0, 1, 102 times 0, 1, 109 times 0) [i] based on linear OA(7³², 33, F₇, 32) (dual of [33, 1, 33]-code or 33-arc in PG(31,7)), using
  - dual of repetition code with length 33 [i]

There is no (76, 108, 573841)-net in base 7, because

the generalized Rao bound for nets shows that 7^mÂ â‰¥ 18 646130 282171 587213 360838 784039 170642 038906 380893 357079 337047 917454 367480 482063 142601 973121 > 7¹⁰⁸ [i]