MinT - Best Known (57−23, 57, s)-Nets in Base 7

Best Known (57−23, 57, s)-Nets in Base 7

Digital (34, 57, 110)-net over F₇, using

1 times m-reduction [i] based on digital (34, 58, 110)-net over F₇, using
- trace code for nets [i] based on digital (5, 29, 55)-net over F₄₉, using
  - net from sequence [i] based on digital (5, 54)-sequence over F₄₉, using
    - Niederreiter sequence [i]

Digital (34, 57, 245)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁵⁷, 245, F₇, 23) (dual of [245, 188, 24]-code), using
- 187 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (4, 2, 2, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 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, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 18 times 0) [i] based on linear OA(7²³, 24, F₇, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,7)), using
  - dual of repetition code with length 24 [i]

There is no (34, 57, 16406)-net in base 7, because

1 times m-reduction [i] would yield (34, 56, 16406)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 211605 480418 193904 919772 996193 400991 630150 937721 > 7⁵⁶ [i]