MinT - Best Known (2, 2+11, s)-Nets in Base 7

Best Known (2, 2+11, s)-Nets in Base 7

(2, 2+11, 13)-Net over F₇ — Constructive and digital

Digital (2, 13, 13)-net over F₇, using

t-expansion [i] based on digital (1, 13, 13)-net over F₇, using
- a shift-net [i]

(2, 2+11, 16)-Net over F₇ — Digital

Digital (2, 13, 16)-net over F₇, using

net from sequence [i] based on digital (2, 15)-sequence over F₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 2 and N(F) ≥ 16, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(2, 2+11, 35)-Net over F₇ — Upper bound on s (digital)

There is no digital (2, 13, 36)-net over F₇, because

1 times m-reduction [i] would yield digital (2, 12, 36)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7¹², 36, F₇, 10) (dual of [36, 24, 11]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(7¹¹, 15, F₇, 10) (dual of [15, 4, 11]-code), but
      - â€œBPMâ€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(7²⁴, 36, F₇, 21) (dual of [36, 12, 22]-code), but
      - discarding factors / shortening the dual code would yield linear OA(7²⁴, 31, F₇, 21) (dual of [31, 7, 22]-code), but
        residual code [i] would yield OA(7³, 9, S₇, 3), but
        bound for OAs with index unity [i]

(2, 2+11, 43)-Net in Base 7 — Upper bound on s

There is no (2, 13, 44)-net in base 7, because

1 times m-reduction [i] would yield (2, 12, 44)-net in base 7, but
- the generalized Rao bound for nets shows that 7^mÂ â‰¥ 14503 827625 > 7¹² [i]