Best Known (18, 18+7, s)-Nets in Base 8
(18, 18+7, 1366)-Net over F8 — Constructive and digital
Digital (18, 25, 1366)-net over F8, using
- net defined by OOA [i] based on linear OOA(825, 1366, F8, 7, 7) (dual of [(1366, 7), 9537, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(825, 4099, F8, 7) (dual of [4099, 4074, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(825, 4100, F8, 7) (dual of [4100, 4075, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(825, 4096, F8, 7) (dual of [4096, 4071, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(821, 4096, F8, 6) (dual of [4096, 4075, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(825, 4100, F8, 7) (dual of [4100, 4075, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(825, 4099, F8, 7) (dual of [4099, 4074, 8]-code), using
(18, 18+7, 4100)-Net over F8 — Digital
Digital (18, 25, 4100)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(825, 4100, F8, 7) (dual of [4100, 4075, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(825, 4096, F8, 7) (dual of [4096, 4071, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(821, 4096, F8, 6) (dual of [4096, 4075, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
(18, 18+7, 4355173)-Net in Base 8 — Upper bound on s
There is no (18, 25, 4355174)-net in base 8, because
- 1 times m-reduction [i] would yield (18, 24, 4355174)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 4722 368023 834123 281830 > 824 [i]