Best Known (26−20, 26, s)-Nets in Base 5
(26−20, 26, 21)-Net over F5 — Constructive and digital
Digital (6, 26, 21)-net over F5, using
- net from sequence [i] based on digital (6, 20)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 6 and N(F) ≥ 21, using
(26−20, 26, 63)-Net over F5 — Upper bound on s (digital)
There is no digital (6, 26, 64)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(526, 64, F5, 20) (dual of [64, 38, 21]-code), but
- construction Y1 [i] would yield
- linear OA(525, 34, F5, 20) (dual of [34, 9, 21]-code), but
- residual code [i] would yield linear OA(55, 13, F5, 4) (dual of [13, 8, 5]-code), but
- “Bou†bound on codes from Brouwer’s database [i]
- residual code [i] would yield linear OA(55, 13, F5, 4) (dual of [13, 8, 5]-code), but
- linear OA(538, 64, F5, 30) (dual of [64, 26, 31]-code), but
- discarding factors / shortening the dual code would yield linear OA(538, 61, F5, 30) (dual of [61, 23, 31]-code), but
- construction Y1 [i] would yield
- linear OA(537, 43, F5, 30) (dual of [43, 6, 31]-code), but
- residual code [i] would yield linear OA(57, 12, F5, 6) (dual of [12, 5, 7]-code), but
- OA(523, 61, S5, 18), but
- the linear programming bound shows that M ≥ 6321 833259 448409 889221 191406 250000 / 502952 782605 714137 > 523 [i]
- linear OA(537, 43, F5, 30) (dual of [43, 6, 31]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(538, 61, F5, 30) (dual of [61, 23, 31]-code), but
- linear OA(525, 34, F5, 20) (dual of [34, 9, 21]-code), but
- construction Y1 [i] would yield
(26−20, 26, 67)-Net in Base 5 — Upper bound on s
There is no (6, 26, 68)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 1 577191 765855 239873 > 526 [i]