Best Known (232−18, 232, s)-Nets in Base 3
(232−18, 232, 1062887)-Net over F3 — Constructive and digital
Digital (214, 232, 1062887)-net over F3, using
- net defined by OOA [i] based on linear OOA(3232, 1062887, F3, 21, 18) (dual of [(1062887, 21), 22320395, 19]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3232, 3188662, F3, 3, 18) (dual of [(3188662, 3), 9565754, 19]-NRT-code), using
- trace code [i] based on linear OOA(9116, 1594331, F9, 3, 18) (dual of [(1594331, 3), 4782877, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(9116, 4782993, F9, 18) (dual of [4782993, 4782877, 19]-code), using
- 1 times truncation [i] based on linear OA(9117, 4782994, F9, 19) (dual of [4782994, 4782877, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(94, 25, F9, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,9)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(9117, 4782994, F9, 19) (dual of [4782994, 4782877, 20]-code), using
- OOA 3-folding [i] based on linear OA(9116, 4782993, F9, 18) (dual of [4782993, 4782877, 19]-code), using
- trace code [i] based on linear OOA(9116, 1594331, F9, 3, 18) (dual of [(1594331, 3), 4782877, 19]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3232, 3188662, F3, 3, 18) (dual of [(3188662, 3), 9565754, 19]-NRT-code), using
(232−18, 232, large)-Net over F3 — Digital
Digital (214, 232, large)-net over F3, using
- 34 times duplication [i] based on digital (210, 228, large)-net over F3, using
- t-expansion [i] based on digital (209, 228, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3228, large, F3, 19) (dual of [large, large−228, 20]-code), using
- 47 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 47 times code embedding in larger space [i] based on linear OA(3181, large, F3, 19) (dual of [large, large−181, 20]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3228, large, F3, 19) (dual of [large, large−228, 20]-code), using
- t-expansion [i] based on digital (209, 228, large)-net over F3, using
(232−18, 232, large)-Net in Base 3 — Upper bound on s
There is no (214, 232, large)-net in base 3, because
- 16 times m-reduction [i] would yield (214, 216, large)-net in base 3, but