Best Known (79−75, 79, s)-Nets in Base 9
(79−75, 79, 30)-Net over F9 — Constructive and digital
Digital (4, 79, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
(79−75, 79, 44)-Net over F9 — Upper bound on s (digital)
There is no digital (4, 79, 45)-net over F9, because
- 39 times m-reduction [i] would yield digital (4, 40, 45)-net over F9, but
- extracting embedded orthogonal array [i] would yield linear OA(940, 45, F9, 36) (dual of [45, 5, 37]-code), but
- construction Y1 [i] would yield
- OA(939, 41, S9, 36), but
- the (dual) Plotkin bound shows that M ≥ 739 044147 071729 616580 416051 031916 488005 / 37 > 939 [i]
- OA(95, 45, S9, 4), but
- discarding factors would yield OA(95, 44, S9, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 60897 > 95 [i]
- discarding factors would yield OA(95, 44, S9, 4), but
- OA(939, 41, S9, 36), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(940, 45, F9, 36) (dual of [45, 5, 37]-code), but
(79−75, 79, 49)-Net in Base 9 — Upper bound on s
There is no (4, 79, 50)-net in base 9, because
- 36 times m-reduction [i] would yield (4, 43, 50)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(943, 50, S9, 39), but
- the linear programming bound shows that M ≥ 307 418272 342644 982111 952555 666013 618532 571037 / 2068 > 943 [i]
- extracting embedded orthogonal array [i] would yield OA(943, 50, S9, 39), but