Best Known (22−19, 22, s)-Nets in Base 7
(22−19, 22, 11)-Net over F7 — Constructive and digital
Digital (3, 22, 11)-net over F7, using
- net from sequence [i] based on digital (3, 10)-sequence over F7, using
(22−19, 22, 20)-Net over F7 — Digital
Digital (3, 22, 20)-net over F7, using
- net from sequence [i] based on digital (3, 19)-sequence over F7, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 3 and N(F) ≥ 20, using
(22−19, 22, 34)-Net over F7 — Upper bound on s (digital)
There is no digital (3, 22, 35)-net over F7, because
- 1 times m-reduction [i] would yield digital (3, 21, 35)-net over F7, but
- extracting embedded orthogonal array [i] would yield linear OA(721, 35, F7, 18) (dual of [35, 14, 19]-code), but
- construction Y1 [i] would yield
- linear OA(720, 23, F7, 18) (dual of [23, 3, 19]-code), but
- “Hi4†bound on codes from Brouwer’s database [i]
- linear OA(714, 35, F7, 12) (dual of [35, 21, 13]-code), but
- discarding factors / shortening the dual code would yield linear OA(714, 30, F7, 12) (dual of [30, 16, 13]-code), but
- construction Y1 [i] would yield
- linear OA(713, 16, F7, 12) (dual of [16, 3, 13]-code), but
- linear OA(716, 30, F7, 14) (dual of [30, 14, 15]-code), but
- discarding factors / shortening the dual code would yield linear OA(716, 24, F7, 14) (dual of [24, 8, 15]-code), but
- residual code [i] would yield OA(72, 9, S7, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 55 > 72 [i]
- residual code [i] would yield OA(72, 9, S7, 2), but
- discarding factors / shortening the dual code would yield linear OA(716, 24, F7, 14) (dual of [24, 8, 15]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(714, 30, F7, 12) (dual of [30, 16, 13]-code), but
- linear OA(720, 23, F7, 18) (dual of [23, 3, 19]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(721, 35, F7, 18) (dual of [35, 14, 19]-code), but
(22−19, 22, 57)-Net in Base 7 — Upper bound on s
There is no (3, 22, 58)-net in base 7, because
- extracting embedded orthogonal array [i] would yield OA(722, 58, S7, 19), but
- the linear programming bound shows that M ≥ 156767 186550 922850 274524 913852 / 36725 317955 > 722 [i]