Best Known (169−19, 169, s)-Nets in Base 3
(169−19, 169, 531441)-Net over F3 — Constructive and digital
Digital (150, 169, 531441)-net over F3, using
- net defined by OOA [i] based on linear OOA(3169, 531441, F3, 19, 19) (dual of [(531441, 19), 10097210, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3169, 4782970, F3, 19) (dual of [4782970, 4782801, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(3169, 4782970, F3, 19) (dual of [4782970, 4782801, 20]-code), using
(169−19, 169, 1195742)-Net over F3 — Digital
Digital (150, 169, 1195742)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3169, 1195742, F3, 4, 19) (dual of [(1195742, 4), 4782799, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3169, 4782968, F3, 19) (dual of [4782968, 4782799, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 4782969, F3, 19) (dual of [4782969, 4782800, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(3169, 4782969, F3, 19) (dual of [4782969, 4782800, 20]-code), using
- OOA 4-folding [i] based on linear OA(3169, 4782968, F3, 19) (dual of [4782968, 4782799, 20]-code), using
(169−19, 169, large)-Net in Base 3 — Upper bound on s
There is no (150, 169, large)-net in base 3, because
- 17 times m-reduction [i] would yield (150, 152, large)-net in base 3, but