Best Known (87−13, 87, s)-Nets in Base 3
(87−13, 87, 9845)-Net over F3 — Constructive and digital
Digital (74, 87, 9845)-net over F3, using
- 32 times duplication [i] based on digital (72, 85, 9845)-net over F3, using
- net defined by OOA [i] based on linear OOA(385, 9845, F3, 13, 13) (dual of [(9845, 13), 127900, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(385, 59071, F3, 13) (dual of [59071, 58986, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(385, 59073, F3, 13) (dual of [59073, 58988, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(361, 59049, F3, 10) (dual of [59049, 58988, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 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(385, 59073, F3, 13) (dual of [59073, 58988, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(385, 59071, F3, 13) (dual of [59071, 58986, 14]-code), using
- net defined by OOA [i] based on linear OOA(385, 9845, F3, 13, 13) (dual of [(9845, 13), 127900, 14]-NRT-code), using
(87−13, 87, 25723)-Net over F3 — Digital
Digital (74, 87, 25723)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(387, 25723, F3, 2, 13) (dual of [(25723, 2), 51359, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(387, 29538, F3, 2, 13) (dual of [(29538, 2), 58989, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(387, 59076, F3, 13) (dual of [59076, 58989, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(381, 59050, F3, 13) (dual of [59050, 58969, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(361, 59050, F3, 9) (dual of [59050, 58989, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(36, 26, F3, 3) (dual of [26, 20, 4]-code or 26-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 2-folding [i] based on linear OA(387, 59076, F3, 13) (dual of [59076, 58989, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(387, 29538, F3, 2, 13) (dual of [(29538, 2), 58989, 14]-NRT-code), using
(87−13, 87, large)-Net in Base 3 — Upper bound on s
There is no (74, 87, large)-net in base 3, because
- 11 times m-reduction [i] would yield (74, 76, large)-net in base 3, but