Best Known (5, 5+39, s)-Nets in Base 8
(5, 5+39, 28)-Net over F8 — Constructive and digital
Digital (5, 44, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
(5, 5+39, 29)-Net over F8 — Digital
Digital (5, 44, 29)-net over F8, using
- net from sequence [i] based on digital (5, 28)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 29, using
(5, 5+39, 52)-Net over F8 — Upper bound on s (digital)
There is no digital (5, 44, 53)-net over F8, because
- 3 times m-reduction [i] would yield digital (5, 41, 53)-net over F8, but
- extracting embedded orthogonal array [i] would yield linear OA(841, 53, F8, 36) (dual of [53, 12, 37]-code), but
- construction Y1 [i] would yield
- linear OA(840, 43, F8, 36) (dual of [43, 3, 37]-code), but
- “Hi4†bound on codes from Brouwer’s database [i]
- linear OA(812, 53, F8, 10) (dual of [53, 41, 11]-code), but
- discarding factors / shortening the dual code would yield linear OA(812, 48, F8, 10) (dual of [48, 36, 11]-code), but
- construction Y1 [i] would yield
- linear OA(811, 16, F8, 10) (dual of [16, 5, 11]-code), but
- “MPa†bound on codes from Brouwer’s database [i]
- linear OA(836, 48, F8, 32) (dual of [48, 12, 33]-code), but
- discarding factors / shortening the dual code would yield linear OA(836, 43, F8, 32) (dual of [43, 7, 33]-code), but
- residual code [i] would yield OA(84, 10, S8, 4), but
- discarding factors / shortening the dual code would yield linear OA(836, 43, F8, 32) (dual of [43, 7, 33]-code), but
- linear OA(811, 16, F8, 10) (dual of [16, 5, 11]-code), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(812, 48, F8, 10) (dual of [48, 36, 11]-code), but
- linear OA(840, 43, F8, 36) (dual of [43, 3, 37]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(841, 53, F8, 36) (dual of [53, 12, 37]-code), but
(5, 5+39, 57)-Net in Base 8 — Upper bound on s
There is no (5, 44, 58)-net in base 8, because
- extracting embedded orthogonal array [i] would yield OA(844, 58, S8, 39), but
- the linear programming bound shows that M ≥ 286 421973 685946 503299 055429 653036 995404 198353 829888 / 43991 122275 > 844 [i]