Best Known (41, 56, s)-Nets in Base 8
(41, 56, 587)-Net over F8 — Constructive and digital
Digital (41, 56, 587)-net over F8, using
- net defined by OOA [i] based on linear OOA(856, 587, F8, 15, 15) (dual of [(587, 15), 8749, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(856, 4110, F8, 15) (dual of [4110, 4054, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(856, 4111, F8, 15) (dual of [4111, 4055, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- 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(841, 4096, F8, 12) (dual of [4096, 4055, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(83, 15, F8, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(856, 4111, F8, 15) (dual of [4111, 4055, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(856, 4110, F8, 15) (dual of [4110, 4054, 16]-code), using
(41, 56, 674)-Net in Base 8 — Constructive
(41, 56, 674)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (9, 16, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 8, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 8, 80)-net over F64, using
- (25, 40, 514)-net in base 8, using
- base change [i] based on digital (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- base change [i] based on digital (15, 30, 514)-net over F16, using
- digital (9, 16, 160)-net over F8, using
(41, 56, 4162)-Net over F8 — Digital
Digital (41, 56, 4162)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(856, 4162, F8, 15) (dual of [4162, 4106, 16]-code), using
- 59 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 51 times 0) [i] based on linear OA(853, 4100, F8, 15) (dual of [4100, 4047, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(13) [i] based on
- 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(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(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(14) ⊂ Ce(13) [i] based on
- 59 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 51 times 0) [i] based on linear OA(853, 4100, F8, 15) (dual of [4100, 4047, 16]-code), using
(41, 56, 6019041)-Net in Base 8 — Upper bound on s
There is no (41, 56, 6019042)-net in base 8, because
- 1 times m-reduction [i] would yield (41, 55, 6019042)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 46 768062 715988 905638 922635 232466 777451 260146 335664 > 855 [i]