Best Known (39−9, 39, s)-Nets in Base 8
(39−9, 39, 8195)-Net over F8 — Constructive and digital
Digital (30, 39, 8195)-net over F8, using
- net defined by OOA [i] based on linear OOA(839, 8195, F8, 9, 9) (dual of [(8195, 9), 73716, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(839, 32781, F8, 9) (dual of [32781, 32742, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(826, 32768, F8, 6) (dual of [32768, 32742, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(83, 13, F8, 2) (dual of [13, 10, 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(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(839, 32781, F8, 9) (dual of [32781, 32742, 10]-code), using
(39−9, 39, 32781)-Net over F8 — Digital
Digital (30, 39, 32781)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(839, 32781, F8, 9) (dual of [32781, 32742, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(826, 32768, F8, 6) (dual of [32768, 32742, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(83, 13, F8, 2) (dual of [13, 10, 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(8) ⊂ Ce(5) [i] based on
(39−9, 39, large)-Net in Base 8 — Upper bound on s
There is no (30, 39, large)-net in base 8, because
- 7 times m-reduction [i] would yield (30, 32, large)-net in base 8, but