Best Known (138−27, 138, s)-Nets in Base 8
(138−27, 138, 2565)-Net over F8 — Constructive and digital
Digital (111, 138, 2565)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (9, 22, 45)-net over F8, using
- net from sequence [i] based on digital (9, 44)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 9 and N(F) ≥ 45, using
- net from sequence [i] based on digital (9, 44)-sequence over F8, using
- digital (89, 116, 2520)-net over F8, using
- net defined by OOA [i] based on linear OOA(8116, 2520, F8, 27, 27) (dual of [(2520, 27), 67924, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8116, 32761, F8, 27) (dual of [32761, 32645, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8116, 32768, F8, 27) (dual of [32768, 32652, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(8116, 32768, F8, 27) (dual of [32768, 32652, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8116, 32761, F8, 27) (dual of [32761, 32645, 28]-code), using
- net defined by OOA [i] based on linear OOA(8116, 2520, F8, 27, 27) (dual of [(2520, 27), 67924, 28]-NRT-code), using
- digital (9, 22, 45)-net over F8, using
(138−27, 138, 5042)-Net in Base 8 — Constructive
(111, 138, 5042)-net in base 8, using
- net defined by OOA [i] based on OOA(8138, 5042, S8, 27, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(8138, 65547, S8, 27), using
- discarding factors based on OA(8138, 65550, S8, 27), using
- discarding parts of the base [i] based on linear OA(16103, 65550, F16, 27) (dual of [65550, 65447, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- linear OA(16101, 65536, F16, 27) (dual of [65536, 65435, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(1689, 65536, F16, 24) (dual of [65536, 65447, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(162, 14, F16, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,16)), using
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- Reed–Solomon code RS(14,16) [i]
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- construction X applied to Ce(26) ⊂ Ce(23) [i] based on
- discarding parts of the base [i] based on linear OA(16103, 65550, F16, 27) (dual of [65550, 65447, 28]-code), using
- discarding factors based on OA(8138, 65550, S8, 27), using
- OOA 13-folding and stacking with additional row [i] based on OA(8138, 65547, S8, 27), using
(138−27, 138, 93665)-Net over F8 — Digital
Digital (111, 138, 93665)-net over F8, using
(138−27, 138, large)-Net in Base 8 — Upper bound on s
There is no (111, 138, large)-net in base 8, because
- 25 times m-reduction [i] would yield (111, 113, large)-net in base 8, but