Best Known (54, 63, s)-Nets in Base 3
(54, 63, 14768)-Net over F3 — Constructive and digital
Digital (54, 63, 14768)-net over F3, using
- net defined by OOA [i] based on linear OOA(363, 14768, F3, 9, 9) (dual of [(14768, 9), 132849, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(363, 59073, F3, 9) (dual of [59073, 59010, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(362, 59072, F3, 9) (dual of [59072, 59010, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 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(341, 59050, F3, 7) (dual of [59050, 59009, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 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 C([0,4]) ⊂ C([0,3]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(362, 59072, F3, 9) (dual of [59072, 59010, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(363, 59073, F3, 9) (dual of [59073, 59010, 10]-code), using
(54, 63, 29536)-Net over F3 — Digital
Digital (54, 63, 29536)-net over F3, using
- 31 times duplication [i] based on digital (53, 62, 29536)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(362, 29536, F3, 2, 9) (dual of [(29536, 2), 59010, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(362, 59072, F3, 9) (dual of [59072, 59010, 10]-code), using
- construction X4 applied to C([0,4]) ⊂ C([0,3]) [i] based on
- 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(341, 59050, F3, 7) (dual of [59050, 59009, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(321, 22, F3, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,3)), using
- dual of repetition code with length 22 [i]
- linear OA(31, 22, F3, 1) (dual of [22, 21, 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 C([0,4]) ⊂ C([0,3]) [i] based on
- OOA 2-folding [i] based on linear OA(362, 59072, F3, 9) (dual of [59072, 59010, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(362, 29536, F3, 2, 9) (dual of [(29536, 2), 59010, 10]-NRT-code), using
(54, 63, large)-Net in Base 3 — Upper bound on s
There is no (54, 63, large)-net in base 3, because
- 7 times m-reduction [i] would yield (54, 56, large)-net in base 3, but