Best Known (44, 44+16, s)-Nets in Base 8
(44, 44+16, 513)-Net over F8 — Constructive and digital
Digital (44, 60, 513)-net over F8, using
- 81 times duplication [i] based on digital (43, 59, 513)-net over F8, using
- t-expansion [i] based on digital (42, 59, 513)-net over F8, using
- net defined by OOA [i] based on linear OOA(859, 513, F8, 17, 17) (dual of [(513, 17), 8662, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(859, 4105, F8, 17) (dual of [4105, 4046, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(857, 4096, F8, 17) (dual of [4096, 4039, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(849, 4096, F8, 14) (dual of [4096, 4047, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(82, 9, F8, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
- extended Reed–Solomon code RSe(7,8) [i]
- Hamming code H(2,8) [i]
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(859, 4105, F8, 17) (dual of [4105, 4046, 18]-code), using
- net defined by OOA [i] based on linear OOA(859, 513, F8, 17, 17) (dual of [(513, 17), 8662, 18]-NRT-code), using
- t-expansion [i] based on digital (42, 59, 513)-net over F8, using
(44, 44+16, 644)-Net in Base 8 — Constructive
(44, 60, 644)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (8, 16, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 8, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 8, 65)-net over F64, using
- (28, 44, 514)-net in base 8, using
- base change [i] based on digital (17, 33, 514)-net over F16, using
- 1 times m-reduction [i] based on digital (17, 34, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 17, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 17, 257)-net over F256, using
- 1 times m-reduction [i] based on digital (17, 34, 514)-net over F16, using
- base change [i] based on digital (17, 33, 514)-net over F16, using
- digital (8, 16, 130)-net over F8, using
(44, 44+16, 4261)-Net over F8 — Digital
Digital (44, 60, 4261)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(860, 4261, F8, 16) (dual of [4261, 4201, 17]-code), using
- 161 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 35 times 0, 1, 117 times 0) [i] based on linear OA(856, 4096, F8, 16) (dual of [4096, 4040, 17]-code), using
- 1 times truncation [i] based on linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using
- 161 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 35 times 0, 1, 117 times 0) [i] based on linear OA(856, 4096, F8, 16) (dual of [4096, 4040, 17]-code), using
(44, 44+16, 3189820)-Net in Base 8 — Upper bound on s
There is no (44, 60, 3189821)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 1 532496 024746 931559 231462 625564 117792 774879 936231 730157 > 860 [i]