Best Known (164−24, 164, s)-Nets in Base 3
(164−24, 164, 4922)-Net over F3 — Constructive and digital
Digital (140, 164, 4922)-net over F3, using
- 1 times m-reduction [i] based on digital (140, 165, 4922)-net over F3, using
- net defined by OOA [i] based on linear OOA(3165, 4922, F3, 25, 25) (dual of [(4922, 25), 122885, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3165, 59065, F3, 25) (dual of [59065, 58900, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3165, 59073, F3, 25) (dual of [59073, 58908, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- 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(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3165, 59073, F3, 25) (dual of [59073, 58908, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3165, 59065, F3, 25) (dual of [59065, 58900, 26]-code), using
- net defined by OOA [i] based on linear OOA(3165, 4922, F3, 25, 25) (dual of [(4922, 25), 122885, 26]-NRT-code), using
(164−24, 164, 20782)-Net over F3 — Digital
Digital (140, 164, 20782)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3164, 20782, F3, 2, 24) (dual of [(20782, 2), 41400, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3164, 29536, F3, 2, 24) (dual of [(29536, 2), 58908, 25]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3162, 29535, F3, 2, 24) (dual of [(29535, 2), 58908, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3162, 59070, F3, 24) (dual of [59070, 58908, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- 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(3141, 59049, F3, 22) (dual of [59049, 58908, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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 Ce(24) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3162, 59070, F3, 24) (dual of [59070, 58908, 25]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3162, 29535, F3, 2, 24) (dual of [(29535, 2), 58908, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3164, 29536, F3, 2, 24) (dual of [(29536, 2), 58908, 25]-NRT-code), using
(164−24, 164, large)-Net in Base 3 — Upper bound on s
There is no (140, 164, large)-net in base 3, because
- 22 times m-reduction [i] would yield (140, 142, large)-net in base 3, but