Best Known (185−27, 185, s)-Nets in Base 3
(185−27, 185, 4544)-Net over F3 — Constructive and digital
Digital (158, 185, 4544)-net over F3, using
- 31 times duplication [i] based on digital (157, 184, 4544)-net over F3, using
- net defined by OOA [i] based on linear OOA(3184, 4544, F3, 27, 27) (dual of [(4544, 27), 122504, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3184, 59073, F3, 27) (dual of [59073, 58889, 28]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3182, 59071, F3, 27) (dual of [59071, 58889, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3181, 59050, F3, 27) (dual of [59050, 58869, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(31, 21, F3, 1) (dual of [21, 20, 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 C([0,13]) ⊂ C([0,12]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3182, 59071, F3, 27) (dual of [59071, 58889, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3184, 59073, F3, 27) (dual of [59073, 58889, 28]-code), using
- net defined by OOA [i] based on linear OOA(3184, 4544, F3, 27, 27) (dual of [(4544, 27), 122504, 28]-NRT-code), using
(185−27, 185, 21277)-Net over F3 — Digital
Digital (158, 185, 21277)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3185, 21277, F3, 2, 27) (dual of [(21277, 2), 42369, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3185, 29537, F3, 2, 27) (dual of [(29537, 2), 58889, 28]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3183, 29536, F3, 2, 27) (dual of [(29536, 2), 58889, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3183, 59072, F3, 27) (dual of [59072, 58889, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3182, 59071, F3, 27) (dual of [59071, 58889, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3181, 59050, F3, 27) (dual of [59050, 58869, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(31, 21, F3, 1) (dual of [21, 20, 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 C([0,13]) ⊂ C([0,12]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3182, 59071, F3, 27) (dual of [59071, 58889, 28]-code), using
- OOA 2-folding [i] based on linear OA(3183, 59072, F3, 27) (dual of [59072, 58889, 28]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3183, 29536, F3, 2, 27) (dual of [(29536, 2), 58889, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3185, 29537, F3, 2, 27) (dual of [(29537, 2), 58889, 28]-NRT-code), using
(185−27, 185, large)-Net in Base 3 — Upper bound on s
There is no (158, 185, large)-net in base 3, because
- 25 times m-reduction [i] would yield (158, 160, large)-net in base 3, but