MinT - Best Known (24, 24+78, s)-Nets in Base 3

Best Known (24, 24+78, s)-Nets in Base 3

(24, 24+78, 32)-Net over F₃ — Constructive and digital

Digital (24, 102, 32)-net over F₃, using

t-expansion [i] based on digital (21, 102, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(24, 24+78, 80)-Net over F₃ — Upper bound on s (digital)

There is no digital (24, 102, 81)-net over F₃, because

27 times m-reduction [i] would yield digital (24, 75, 81)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁷⁵, 81, F₃, 51) (dual of [81, 6, 52]-code), but
  - â€œMaâ€ bound on codes from Brouwerâ€™s database [i]

(24, 24+78, 81)-Net in Base 3 — Upper bound on s

There is no (24, 102, 82)-net in base 3, because

28 times m-reduction [i] would yield (24, 74, 82)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁷⁴, 82, S₃, 50), but
  - the linear programming bound shows that MÂ â‰¥ 1133 201025 509985 412089 971278 248938 614941 / 5015 > 3⁷⁴ [i]