MinT - Best Known (63−55, 63, s)-Nets in Base 7

Best Known (63−55, 63, s)-Nets in Base 7

(63−55, 63, 16)-Net over F₇ — Constructive and digital

Digital (8, 63, 16)-net over F₇, using

net from sequence [i] based on digital (8, 15)-sequence over F₇, using
- Niederreiter sequence [i]

(63−55, 63, 32)-Net over F₇ — Digital

Digital (8, 63, 32)-net over F₇, using

t-expansion [i] based on digital (7, 63, 32)-net over F₇, using
- net from sequence [i] based on digital (7, 31)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 7 and N(F) ≥ 32, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(63−55, 63, 63)-Net over F₇ — Upper bound on s (digital)

There is no digital (8, 63, 64)-net over F₇, because

6 times m-reduction [i] would yield digital (8, 57, 64)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7⁵⁷, 64, F₇, 49) (dual of [64, 7, 50]-code), but
  - residual code [i] would yield linear OA(7⁸, 14, F₇, 7) (dual of [14, 6, 8]-code), but
    - â€œMPaâ€ bound on codes from Brouwerâ€™s database [i]

(63−55, 63, 66)-Net in Base 7 — Upper bound on s

There is no (8, 63, 67)-net in base 7, because

6 times m-reduction [i] would yield (8, 57, 67)-net in base 7, but
- extracting embedded orthogonal array [i] would yield OA(7⁵⁷, 67, S₇, 49), but
  - the linear programming bound shows that MÂ â‰¥ 14 690613 296110 757318 840379 593851 330929 325424 135975 451203 / 9 440000 > 7⁵⁷ [i]