MinT - Best Known (82−76, 82, s)-Nets in Base 7

Best Known (82−76, 82, s)-Nets in Base 7

(82−76, 82, 14)-Net over F₇ — Constructive and digital

Digital (6, 82, 14)-net over F₇, using

net from sequence [i] based on digital (6, 13)-sequence over F₇, using
- Niederreiter sequence [i]

(82−76, 82, 24)-Net over F₇ — Digital

Digital (6, 82, 24)-net over F₇, using

t-expansion [i] based on digital (4, 82, 24)-net over F₇, using
- net from sequence [i] based on digital (4, 23)-sequence over F₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₇ with g(F) = 4 and N(F) ≥ 24, using
    - an algebraic function field over â„¤₇ by Yeo [i]

(82−76, 82, 51)-Net over F₇ — Upper bound on s (digital)

There is no digital (6, 82, 52)-net over F₇, because

34 times m-reduction [i] would yield digital (6, 48, 52)-net over F₇, but
- extracting embedded orthogonal array [i] would yield linear OA(7⁴⁸, 52, F₇, 42) (dual of [52, 4, 43]-code), but
  - residual code [i] would yield linear OA(7⁶, 9, F₇, 6) (dual of [9, 3, 7]-code or 9-arc in PG(5,7)), but
    - bound for MDS codes [i]

(82−76, 82, 52)-Net in Base 7 — Upper bound on s

There is no (6, 82, 53)-net in base 7, because

34 times m-reduction [i] would yield (6, 48, 53)-net in base 7, but
- extracting embedded orthogonal array [i] would yield OA(7⁴⁸, 53, S₇, 42), but
  - the linear programming bound shows that MÂ â‰¥ 1 688354 937995 529770 296589 707473 548368 684846 / 43 > 7⁴⁸ [i]