Best Known (155, 155+27, s)-Nets in Base 3
(155, 155+27, 4543)-Net over F3 — Constructive and digital
Digital (155, 182, 4543)-net over F3, using
- net defined by OOA [i] based on linear OOA(3182, 4543, F3, 27, 27) (dual of [(4543, 27), 122479, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3182, 59060, F3, 27) (dual of [59060, 58878, 28]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code 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(3182, 59060, F3, 27) (dual of [59060, 58878, 28]-code), using
(155, 155+27, 19690)-Net over F3 — Digital
Digital (155, 182, 19690)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3182, 19690, F3, 3, 27) (dual of [(19690, 3), 58888, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3182, 59070, F3, 27) (dual of [59070, 58888, 28]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3182, 59071, F3, 27) (dual of [59071, 58889, 28]-code), using
- OOA 3-folding [i] based on linear OA(3182, 59070, F3, 27) (dual of [59070, 58888, 28]-code), using
(155, 155+27, large)-Net in Base 3 — Upper bound on s
There is no (155, 182, large)-net in base 3, because
- 25 times m-reduction [i] would yield (155, 157, large)-net in base 3, but