MinT - Best Known (13−11, 13, s)-Nets in Base 8

Best Known (13−11, 13, s)-Nets in Base 8

(13−11, 13, 17)-Net over F₈ — Constructive and digital

Digital (2, 13, 17)-net over F₈, using

net from sequence [i] based on digital (2, 16)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
  - a function field by SÃ©mirat [i]

(13−11, 13, 18)-Net over F₈ — Digital

Digital (2, 13, 18)-net over F₈, using

net from sequence [i] based on digital (2, 17)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 18, using
  - sharp bound on the number of rational points on curves with genus 2 [i]

(13−11, 13, 47)-Net over F₈ — Upper bound on s (digital)

There is no digital (2, 13, 48)-net over F₈, because

1 times m-reduction [i] would yield digital (2, 12, 48)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8¹², 48, F₈, 10) (dual of [48, 36, 11]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(8¹¹, 16, F₈, 10) (dual of [16, 5, 11]-code), but
      - â€œMPaâ€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(8³⁶, 48, F₈, 32) (dual of [48, 12, 33]-code), but
      - discarding factors / shortening the dual code would yield linear OA(8³⁶, 43, F₈, 32) (dual of [43, 7, 33]-code), but
        residual code [i] would yield OA(8⁴, 10, S₈, 4), but
        bound for OAs with index unity [i]

(13−11, 13, 52)-Net in Base 8 — Upper bound on s

There is no (2, 13, 53)-net in base 8, because

1 times m-reduction [i] would yield (2, 12, 53)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 74872 696088 > 8¹² [i]