Best Known (58−9, 58, s)-Nets in Base 8
(58−9, 58, 2097150)-Net over F8 — Constructive and digital
Digital (49, 58, 2097150)-net over F8, using
- 81 times duplication [i] based on digital (48, 57, 2097150)-net over F8, using
- net defined by OOA [i] based on linear OOA(857, 2097150, F8, 9, 9) (dual of [(2097150, 9), 18874293, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(857, 8388601, F8, 9) (dual of [8388601, 8388544, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(857, 8388601, F8, 9) (dual of [8388601, 8388544, 10]-code), using
- net defined by OOA [i] based on linear OOA(857, 2097150, F8, 9, 9) (dual of [(2097150, 9), 18874293, 10]-NRT-code), using
(58−9, 58, large)-Net over F8 — Digital
Digital (49, 58, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(858, large, F8, 9) (dual of [large, large−58, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times code embedding in larger space [i] based on linear OA(857, large, F8, 9) (dual of [large, large−57, 10]-code), using
(58−9, 58, large)-Net in Base 8 — Upper bound on s
There is no (49, 58, large)-net in base 8, because
- 7 times m-reduction [i] would yield (49, 51, large)-net in base 8, but