Best Known (38−10, 38, s)-Nets in Base 8
(38−10, 38, 1639)-Net over F8 — Constructive and digital
Digital (28, 38, 1639)-net over F8, using
- net defined by OOA [i] based on linear OOA(838, 1639, F8, 10, 10) (dual of [(1639, 10), 16352, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(838, 8195, F8, 10) (dual of [8195, 8157, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(838, 8196, F8, 10) (dual of [8196, 8158, 11]-code), using
- trace code [i] based on linear OA(6419, 4098, F64, 10) (dual of [4098, 4079, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(6417, 4096, F64, 9) (dual of [4096, 4079, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(9) ⊂ Ce(8) [i] based on
- trace code [i] based on linear OA(6419, 4098, F64, 10) (dual of [4098, 4079, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(838, 8196, F8, 10) (dual of [8196, 8158, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(838, 8195, F8, 10) (dual of [8195, 8157, 11]-code), using
(38−10, 38, 8075)-Net over F8 — Digital
Digital (28, 38, 8075)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(838, 8075, F8, 10) (dual of [8075, 8037, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(838, 8192, F8, 10) (dual of [8192, 8154, 11]-code), using
- trace code [i] based on linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- trace code [i] based on linear OA(6419, 4096, F64, 10) (dual of [4096, 4077, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(838, 8192, F8, 10) (dual of [8192, 8154, 11]-code), using
(38−10, 38, 2717826)-Net in Base 8 — Upper bound on s
There is no (28, 38, 2717827)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 20769 219040 022146 402206 252136 268608 > 838 [i]