Best Known (159, 186, s)-Nets in Base 3
(159, 186, 4544)-Net over F3 — Constructive and digital
Digital (159, 186, 4544)-net over F3, using
- 32 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
(159, 186, 22275)-Net over F3 — Digital
Digital (159, 186, 22275)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3186, 22275, F3, 2, 27) (dual of [(22275, 2), 44364, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3186, 29538, F3, 2, 27) (dual of [(29538, 2), 58890, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3186, 59076, F3, 27) (dual of [59076, 58890, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3186, 59077, F3, 27) (dual of [59077, 58891, 28]-code), using
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3161, 59049, F3, 25) (dual of [59049, 58888, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 24, F3, 1) (dual of [24, 23, 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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(27) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3186, 59077, F3, 27) (dual of [59077, 58891, 28]-code), using
- OOA 2-folding [i] based on linear OA(3186, 59076, F3, 27) (dual of [59076, 58890, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(3186, 29538, F3, 2, 27) (dual of [(29538, 2), 58890, 28]-NRT-code), using
(159, 186, large)-Net in Base 3 — Upper bound on s
There is no (159, 186, large)-net in base 3, because
- 25 times m-reduction [i] would yield (159, 161, large)-net in base 3, but