MinT - Best Known (29, 50, s)-Nets in Base 7

Best Known (29, 50, s)-Nets in Base 7

Digital (29, 50, 108)-net over F₇, using

trace code for nets [i] based on digital (4, 25, 54)-net over F₄₉, using
- net from sequence [i] based on digital (4, 53)-sequence over F₄₉, using
  - Niederreiter sequence [i]

Digital (29, 50, 190)-net over F₇, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(7⁵⁰, 190, F₇, 21) (dual of [190, 140, 22]-code), using
- 30 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 10 times 0, 1, 14 times 0) [i] based on linear OA(7⁴⁶, 156, F₇, 21) (dual of [156, 110, 22]-code), using
  - trace code [i] based on linear OA(49²³, 78, F₄₉, 21) (dual of [78, 55, 22]-code), using
    - extended algebraic-geometric code AG_e(F,56P) [i] based on function field F/F₄₉ with g(F) = 2 and N(F) ≥ 78, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

There is no (29, 50, 10437)-net in base 7, because

1 times m-reduction [i] would yield (29, 49, 10437)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 257081 632906 171677 430607 016094 057565 768473 > 7⁴⁹ [i]