Best Known (45, 45+18, s)-Nets in Base 8
(45, 45+18, 456)-Net over F8 — Constructive and digital
Digital (45, 63, 456)-net over F8, using
- net defined by OOA [i] based on linear OOA(863, 456, F8, 18, 18) (dual of [(456, 18), 8145, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(863, 4104, F8, 18) (dual of [4104, 4041, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(863, 4105, F8, 18) (dual of [4105, 4042, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(861, 4096, F8, 18) (dual of [4096, 4035, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(853, 4096, F8, 15) (dual of [4096, 4043, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(863, 4105, F8, 18) (dual of [4105, 4042, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(863, 4104, F8, 18) (dual of [4104, 4041, 19]-code), using
(45, 45+18, 547)-Net in Base 8 — Constructive
(45, 63, 547)-net in base 8, using
- (u, u+v)-construction [i] based on
- (6, 15, 33)-net in base 8, using
- 1 times m-reduction [i] based on (6, 16, 33)-net in base 8, using
- base change [i] based on digital (2, 12, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- base change [i] based on digital (2, 12, 33)-net over F16, using
- 1 times m-reduction [i] based on (6, 16, 33)-net in base 8, using
- (30, 48, 514)-net in base 8, using
- base change [i] based on digital (18, 36, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- base change [i] based on digital (18, 36, 514)-net over F16, using
- (6, 15, 33)-net in base 8, using
(45, 45+18, 3059)-Net over F8 — Digital
Digital (45, 63, 3059)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(863, 3059, F8, 18) (dual of [3059, 2996, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(863, 4105, F8, 18) (dual of [4105, 4042, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(861, 4096, F8, 18) (dual of [4096, 4035, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(853, 4096, F8, 15) (dual of [4096, 4043, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(863, 4105, F8, 18) (dual of [4105, 4042, 19]-code), using
(45, 45+18, 1242457)-Net in Base 8 — Upper bound on s
There is no (45, 63, 1242458)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 784 641738 744953 506867 515131 623970 635210 091710 195636 975705 > 863 [i]