MinT - Best Known (26−23, 26, s)-Nets in Base 7

Best Known (26−23, 26, s)-Nets in Base 7

(26−23, 26, 11)-Net over F₇ — Constructive and digital

Digital (3, 26, 11)-net over F₇, using

net from sequence [i] based on digital (3, 10)-sequence over F₇, using
- Niederreiter sequence [i]

(26−23, 26, 20)-Net over F₇ — Digital

Digital (3, 26, 20)-net over F₇, using

net from sequence [i] based on digital (3, 19)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 3 and N(F) ≥ 20, using
  - fields by Serre with genus 3 or 4 [i]

(26−23, 26, 28)-Net over F₇ — Upper bound on s (digital)

There is no digital (3, 26, 29)-net over F₇, because

2 times m-reduction [i] would yield digital (3, 24, 29)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7²⁴, 29, F₇, 21) (dual of [29, 5, 22]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(7²³, 25, S₇, 21), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 383 162462 761132 828802 / 11 > 7²³ [i]
    2. OA(7⁵, 29, S₇, 4), but
      - the linear programming bound shows that MÂ â‰¥ 3 384381 / 197 > 7⁵ [i]

(26−23, 26, 43)-Net in Base 7 — Upper bound on s

There is no (3, 26, 44)-net in base 7, because

extracting embedded orthogonal array [i] would yield OA(7²⁶, 44, S₇, 23), but
- the linear programming bound shows that MÂ â‰¥ 538 872446 391031 079337 068171 964389 / 54132 822077 > 7²⁶ [i]