Best Known (216−17, 216, s)-Nets in Base 3
(216−17, 216, 1195746)-Net over F3 — Constructive and digital
Digital (199, 216, 1195746)-net over F3, using
- net defined by OOA [i] based on linear OOA(3216, 1195746, F3, 18, 17) (dual of [(1195746, 18), 21523212, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(3216, 4782985, F3, 2, 17) (dual of [(4782985, 2), 9565754, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3216, 4782986, F3, 2, 17) (dual of [(4782986, 2), 9565756, 18]-NRT-code), using
- trace code [i] based on linear OOA(9108, 2391493, F9, 2, 17) (dual of [(2391493, 2), 4782878, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(9108, 4782986, F9, 17) (dual of [4782986, 4782878, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(14) ⊂ Ce(13) [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(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(91, 16, F9, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(9108, 4782986, F9, 17) (dual of [4782986, 4782878, 18]-code), using
- trace code [i] based on linear OOA(9108, 2391493, F9, 2, 17) (dual of [(2391493, 2), 4782878, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3216, 4782986, F3, 2, 17) (dual of [(4782986, 2), 9565756, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(3216, 4782985, F3, 2, 17) (dual of [(4782985, 2), 9565754, 18]-NRT-code), using
(216−17, 216, large)-Net over F3 — Digital
Digital (199, 216, large)-net over F3, using
- t-expansion [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
(216−17, 216, large)-Net in Base 3 — Upper bound on s
There is no (199, 216, large)-net in base 3, because
- 15 times m-reduction [i] would yield (199, 201, large)-net in base 3, but