Best Known (27−22, 27, s)-Nets in Base 5
(27−22, 27, 20)-Net over F5 — Constructive and digital
Digital (5, 27, 20)-net over F5, using
- net from sequence [i] based on digital (5, 19)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 5 and N(F) ≥ 20, using
(27−22, 27, 33)-Net over F5 — Upper bound on s (digital)
There is no digital (5, 27, 34)-net over F5, because
- 2 times m-reduction [i] would yield digital (5, 25, 34)-net over F5, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(525, 34, F5, 20) (dual of [34, 9, 21]-code), but
(27−22, 27, 47)-Net in Base 5 — Upper bound on s
There is no (5, 27, 48)-net in base 5, because
- extracting embedded orthogonal array [i] would yield OA(527, 48, S5, 22), but
- the linear programming bound shows that M ≥ 8 529310 671508 853323 757648 468017 578125 / 1 109510 787428 226381 > 527 [i]