Best Known (14−12, 14, s)-Nets in Base 8
(14−12, 14, 17)-Net over F8 — Constructive and digital
Digital (2, 14, 17)-net over F8, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 17, using
(14−12, 14, 18)-Net over F8 — Digital
Digital (2, 14, 18)-net over F8, using
- net from sequence [i] based on digital (2, 17)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 18, using
(14−12, 14, 31)-Net over F8 — Upper bound on s (digital)
There is no digital (2, 14, 32)-net over F8, because
- extracting embedded orthogonal array [i] would yield linear OA(814, 32, F8, 12) (dual of [32, 18, 13]-code), but
- construction Y1 [i] would yield
- linear OA(813, 16, F8, 12) (dual of [16, 3, 13]-code), but
- “Hi4†bound on codes from Brouwer’s database [i]
- linear OA(818, 32, F8, 16) (dual of [32, 14, 17]-code), but
- discarding factors / shortening the dual code would yield linear OA(818, 27, F8, 16) (dual of [27, 9, 17]-code), but
- residual code [i] would yield OA(82, 10, S8, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 71 > 82 [i]
- residual code [i] would yield OA(82, 10, S8, 2), but
- discarding factors / shortening the dual code would yield linear OA(818, 27, F8, 16) (dual of [27, 9, 17]-code), but
- linear OA(813, 16, F8, 12) (dual of [16, 3, 13]-code), but
- construction Y1 [i] would yield
(14−12, 14, 51)-Net in Base 8 — Upper bound on s
There is no (2, 14, 52)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 4 666928 710900 > 814 [i]