Best Known (215−17, 215, s)-Nets in Base 3
(215−17, 215, 1195745)-Net over F3 — Constructive and digital
Digital (198, 215, 1195745)-net over F3, using
- 31 times duplication [i] based on digital (197, 214, 1195745)-net over F3, using
- net defined by OOA [i] based on linear OOA(3214, 1195745, F3, 18, 17) (dual of [(1195745, 18), 21523196, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(3214, 4782981, F3, 2, 17) (dual of [(4782981, 2), 9565748, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3214, 4782984, F3, 2, 17) (dual of [(4782984, 2), 9565754, 18]-NRT-code), using
- trace code [i] based on linear OOA(9107, 2391492, F9, 2, 17) (dual of [(2391492, 2), 4782877, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(9107, 4782984, F9, 17) (dual of [4782984, 4782877, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(9106, 4782969, F9, 17) (dual of [4782969, 4782863, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(91, 15, F9, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(9107, 4782984, F9, 17) (dual of [4782984, 4782877, 18]-code), using
- trace code [i] based on linear OOA(9107, 2391492, F9, 2, 17) (dual of [(2391492, 2), 4782877, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3214, 4782984, F3, 2, 17) (dual of [(4782984, 2), 9565754, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(3214, 4782981, F3, 2, 17) (dual of [(4782981, 2), 9565748, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3214, 1195745, F3, 18, 17) (dual of [(1195745, 18), 21523196, 18]-NRT-code), using
(215−17, 215, large)-Net over F3 — Digital
Digital (198, 215, large)-net over F3, using
- 1 times m-reduction [i] based on digital (198, 216, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 36 times code embedding in larger space [i] based on linear OA(3180, large, F3, 18) (dual of [large, large−180, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3216, large, F3, 18) (dual of [large, large−216, 19]-code), using
(215−17, 215, large)-Net in Base 3 — Upper bound on s
There is no (198, 215, large)-net in base 3, because
- 15 times m-reduction [i] would yield (198, 200, large)-net in base 3, but