Best Known (66−18, 66, s)-Nets in Base 8
(66−18, 66, 457)-Net over F8 — Constructive and digital
Digital (48, 66, 457)-net over F8, using
- net defined by OOA [i] based on linear OOA(866, 457, F8, 18, 18) (dual of [(457, 18), 8160, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(866, 4113, F8, 18) (dual of [4113, 4047, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(866, 4114, F8, 18) (dual of [4114, 4048, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [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(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(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(84, 17, F8, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,8)), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(866, 4114, F8, 18) (dual of [4114, 4048, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(866, 4113, F8, 18) (dual of [4113, 4047, 19]-code), using
(66−18, 66, 644)-Net in Base 8 — Constructive
(48, 66, 644)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (9, 18, 130)-net over F8, using
- trace code for nets [i] based on digital (0, 9, 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, 9, 65)-net over F64, 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
- digital (9, 18, 130)-net over F8, using
(66−18, 66, 4154)-Net over F8 — Digital
Digital (48, 66, 4154)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(866, 4154, F8, 18) (dual of [4154, 4088, 19]-code), using
- 49 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 10 times 0, 1, 32 times 0) [i] based on linear OA(861, 4100, F8, 18) (dual of [4100, 4039, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [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(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(80, 4, F8, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- 49 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 0, 1, 10 times 0, 1, 32 times 0) [i] based on linear OA(861, 4100, F8, 18) (dual of [4100, 4039, 19]-code), using
(66−18, 66, 2484919)-Net in Base 8 — Upper bound on s
There is no (48, 66, 2484920)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 401734 699511 635562 609021 654214 272563 690881 116172 027961 421774 > 866 [i]