Best Known (164, 164+28, s)-Nets in Base 4
(164, 164+28, 18726)-Net over F4 — Constructive and digital
Digital (164, 192, 18726)-net over F4, using
- net defined by OOA [i] based on linear OOA(4192, 18726, F4, 28, 28) (dual of [(18726, 28), 524136, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(4192, 262164, F4, 28) (dual of [262164, 261972, 29]-code), using
- 1 times truncation [i] based on linear OA(4193, 262165, F4, 29) (dual of [262165, 261972, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- 1 times truncation [i] based on linear OA(4193, 262165, F4, 29) (dual of [262165, 261972, 30]-code), using
- OA 14-folding and stacking [i] based on linear OA(4192, 262164, F4, 28) (dual of [262164, 261972, 29]-code), using
(164, 164+28, 127671)-Net over F4 — Digital
Digital (164, 192, 127671)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4192, 127671, F4, 2, 28) (dual of [(127671, 2), 255150, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4192, 131082, F4, 2, 28) (dual of [(131082, 2), 261972, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4192, 262164, F4, 28) (dual of [262164, 261972, 29]-code), using
- 1 times truncation [i] based on linear OA(4193, 262165, F4, 29) (dual of [262165, 261972, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- 1 times truncation [i] based on linear OA(4193, 262165, F4, 29) (dual of [262165, 261972, 30]-code), using
- OOA 2-folding [i] based on linear OA(4192, 262164, F4, 28) (dual of [262164, 261972, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(4192, 131082, F4, 2, 28) (dual of [(131082, 2), 261972, 29]-NRT-code), using
(164, 164+28, large)-Net in Base 4 — Upper bound on s
There is no (164, 192, large)-net in base 4, because
- 26 times m-reduction [i] would yield (164, 166, large)-net in base 4, but