Best Known (29−9, 29, s)-Nets in Base 8
(29−9, 29, 1023)-Net over F8 — Constructive and digital
Digital (20, 29, 1023)-net over F8, using
- net defined by OOA [i] based on linear OOA(829, 1023, F8, 9, 9) (dual of [(1023, 9), 9178, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(829, 4093, F8, 9) (dual of [4093, 4064, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(829, 4093, F8, 9) (dual of [4093, 4064, 10]-code), using
(29−9, 29, 2048)-Net over F8 — Digital
Digital (20, 29, 2048)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(829, 2048, F8, 2, 9) (dual of [(2048, 2), 4067, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 2-folding [i] based on linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using
(29−9, 29, 663106)-Net in Base 8 — Upper bound on s
There is no (20, 29, 663107)-net in base 8, because
- 1 times m-reduction [i] would yield (20, 28, 663107)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 19 342864 743882 646319 660112 > 828 [i]