Best Known (41, 55, s)-Nets in Base 8
(41, 55, 1171)-Net over F8 — Constructive and digital
Digital (41, 55, 1171)-net over F8, using
- net defined by OOA [i] based on linear OOA(855, 1171, F8, 14, 14) (dual of [(1171, 14), 16339, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(855, 8197, F8, 14) (dual of [8197, 8142, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(854, 8196, F8, 14) (dual of [8196, 8142, 15]-code), using
- trace code [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(854, 8196, F8, 14) (dual of [8196, 8142, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(855, 8197, F8, 14) (dual of [8197, 8142, 15]-code), using
(41, 55, 8198)-Net over F8 — Digital
Digital (41, 55, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(855, 8198, F8, 14) (dual of [8198, 8143, 15]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(854, 8196, F8, 14) (dual of [8196, 8142, 15]-code), using
- trace code [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(6427, 4096, F64, 14) (dual of [4096, 4069, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(6427, 4098, F64, 14) (dual of [4098, 4071, 15]-code), using
- linear OA(854, 8197, F8, 13) (dual of [8197, 8143, 14]-code), using Gilbert–Varšamov bound and bm = 854 > Vbs−1(k−1) = 2 634237 917083 705149 649971 188156 376516 285541 156864 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 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
- linear OA(854, 8196, F8, 14) (dual of [8196, 8142, 15]-code), using
- construction X with Varšamov bound [i] based on
(41, 55, 6019041)-Net in Base 8 — Upper bound on s
There is no (41, 55, 6019042)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 46 768062 715988 905638 922635 232466 777451 260146 335664 > 855 [i]