Best Known (214−23, 214, s)-Nets in Base 3
(214−23, 214, 434816)-Net over F3 — Constructive and digital
Digital (191, 214, 434816)-net over F3, using
- 33 times duplication [i] based on digital (188, 211, 434816)-net over F3, using
- net defined by OOA [i] based on linear OOA(3211, 434816, F3, 23, 23) (dual of [(434816, 23), 10000557, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3211, 4782977, F3, 23) (dual of [4782977, 4782766, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, 4782983, F3, 23) (dual of [4782983, 4782772, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3211, 4782983, F3, 23) (dual of [4782983, 4782772, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3211, 4782977, F3, 23) (dual of [4782977, 4782766, 24]-code), using
- net defined by OOA [i] based on linear OOA(3211, 434816, F3, 23, 23) (dual of [(434816, 23), 10000557, 24]-NRT-code), using
(214−23, 214, 1195746)-Net over F3 — Digital
Digital (191, 214, 1195746)-net over F3, using
- 32 times duplication [i] based on digital (189, 212, 1195746)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3212, 1195746, F3, 4, 23) (dual of [(1195746, 4), 4782772, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3212, 4782984, F3, 23) (dual of [4782984, 4782772, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3211, 4782983, F3, 23) (dual of [4782983, 4782772, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3197, 4782969, F3, 22) (dual of [4782969, 4782772, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 14, F3, 0) (dual of [14, 14, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3211, 4782983, F3, 23) (dual of [4782983, 4782772, 24]-code), using
- OOA 4-folding [i] based on linear OA(3212, 4782984, F3, 23) (dual of [4782984, 4782772, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3212, 1195746, F3, 4, 23) (dual of [(1195746, 4), 4782772, 24]-NRT-code), using
(214−23, 214, large)-Net in Base 3 — Upper bound on s
There is no (191, 214, large)-net in base 3, because
- 21 times m-reduction [i] would yield (191, 193, large)-net in base 3, but