Best Known (115−12, 115, s)-Nets in Base 3
(115−12, 115, 797166)-Net over F3 — Constructive and digital
Digital (103, 115, 797166)-net over F3, using
- 31 times duplication [i] based on digital (102, 114, 797166)-net over F3, using
- net defined by OOA [i] based on linear OOA(3114, 797166, F3, 12, 12) (dual of [(797166, 12), 9565878, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3114, 4782996, F3, 12) (dual of [4782996, 4782882, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3114, 4782998, F3, 12) (dual of [4782998, 4782884, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(3113, 4782969, F3, 13) (dual of [4782969, 4782856, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(385, 4782969, F3, 10) (dual of [4782969, 4782884, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(31, 29, F3, 1) (dual of [29, 28, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(3114, 4782998, F3, 12) (dual of [4782998, 4782884, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(3114, 4782996, F3, 12) (dual of [4782996, 4782882, 13]-code), using
- net defined by OOA [i] based on linear OOA(3114, 797166, F3, 12, 12) (dual of [(797166, 12), 9565878, 13]-NRT-code), using
(115−12, 115, 2028814)-Net over F3 — Digital
Digital (103, 115, 2028814)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3115, 2028814, F3, 2, 12) (dual of [(2028814, 2), 4057513, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3115, 2391500, F3, 2, 12) (dual of [(2391500, 2), 4782885, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3115, 4783000, F3, 12) (dual of [4783000, 4782885, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3114, 4782999, F3, 12) (dual of [4782999, 4782885, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(3113, 4782969, F3, 13) (dual of [4782969, 4782856, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(385, 4782969, F3, 10) (dual of [4782969, 4782884, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(329, 30, F3, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
- dual of repetition code with length 30 [i]
- linear OA(31, 30, F3, 1) (dual of [30, 29, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3114, 4782999, F3, 12) (dual of [4782999, 4782885, 13]-code), using
- OOA 2-folding [i] based on linear OA(3115, 4783000, F3, 12) (dual of [4783000, 4782885, 13]-code), using
- discarding factors / shortening the dual code based on linear OOA(3115, 2391500, F3, 2, 12) (dual of [(2391500, 2), 4782885, 13]-NRT-code), using
(115−12, 115, large)-Net in Base 3 — Upper bound on s
There is no (103, 115, large)-net in base 3, because
- 10 times m-reduction [i] would yield (103, 105, large)-net in base 3, but