MinT - Best Known (3, 31, s)-Nets in Base 7

Best Known (3, 31, s)-Nets in Base 7

(3, 31, 11)-Net over F₇ — Constructive and digital

Digital (3, 31, 11)-net over F₇, using

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

(3, 31, 20)-Net over F₇ — Digital

Digital (3, 31, 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]

(3, 31, 28)-Net over F₇ — Upper bound on s (digital)

There is no digital (3, 31, 29)-net over F₇, because

7 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]

(3, 31, 32)-Net in Base 7 — Upper bound on s

There is no (3, 31, 33)-net in base 7, because

extracting embedded orthogonal array [i] would yield OA(7³¹, 33, S₇, 28), but
- the (dual) Plotkin bound shows that MÂ â‰¥ 5522 138371 219603 231526 496005 / 29 > 7³¹ [i]