Best Known (97, 97+12, s)-Nets in Base 3
(97, 97+12, 265725)-Net over F3 — Constructive and digital
Digital (97, 109, 265725)-net over F3, using
- t-expansion [i] based on digital (96, 109, 265725)-net over F3, using
- net defined by OOA [i] based on linear OOA(3109, 265725, F3, 13, 13) (dual of [(265725, 13), 3454316, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3109, 1594351, F3, 13) (dual of [1594351, 1594242, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3109, 1594353, F3, 13) (dual of [1594353, 1594244, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(3109, 1594353, F3, 13) (dual of [1594353, 1594244, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3109, 1594351, F3, 13) (dual of [1594351, 1594242, 14]-code), using
- net defined by OOA [i] based on linear OOA(3109, 265725, F3, 13, 13) (dual of [(265725, 13), 3454316, 14]-NRT-code), using
(97, 97+12, 797177)-Net over F3 — Digital
Digital (97, 109, 797177)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3109, 797177, F3, 2, 12) (dual of [(797177, 2), 1594245, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3107, 797176, F3, 2, 12) (dual of [(797176, 2), 1594245, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3107, 1594352, F3, 12) (dual of [1594352, 1594245, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3106, 1594351, F3, 12) (dual of [1594351, 1594245, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(3105, 1594323, F3, 13) (dual of [1594323, 1594218, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(327, 28, F3, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
- dual of repetition code with length 28 [i]
- linear OA(31, 28, F3, 1) (dual of [28, 27, 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(3106, 1594351, F3, 12) (dual of [1594351, 1594245, 13]-code), using
- OOA 2-folding [i] based on linear OA(3107, 1594352, F3, 12) (dual of [1594352, 1594245, 13]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3107, 797176, F3, 2, 12) (dual of [(797176, 2), 1594245, 13]-NRT-code), using
(97, 97+12, large)-Net in Base 3 — Upper bound on s
There is no (97, 109, large)-net in base 3, because
- 10 times m-reduction [i] would yield (97, 99, large)-net in base 3, but