Best Known (98, 98+13, s)-Nets in Base 3
(98, 98+13, 265725)-Net over F3 — Constructive and digital
Digital (98, 111, 265725)-net over F3, using
- 32 times duplication [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
(98, 98+13, 531452)-Net over F3 — Digital
Digital (98, 111, 531452)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3111, 531452, F3, 3, 13) (dual of [(531452, 3), 1594245, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3111, 1594356, F3, 13) (dual of [1594356, 1594245, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(3105, 1594324, F3, 13) (dual of [1594324, 1594219, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(379, 1594324, F3, 9) (dual of [1594324, 1594245, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(36, 32, F3, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- OOA 3-folding [i] based on linear OA(3111, 1594356, F3, 13) (dual of [1594356, 1594245, 14]-code), using
(98, 98+13, large)-Net in Base 3 — Upper bound on s
There is no (98, 111, large)-net in base 3, because
- 11 times m-reduction [i] would yield (98, 100, large)-net in base 3, but