Best Known (166−24, 166, s)-Nets in Base 4
(166−24, 166, 21847)-Net over F4 — Constructive and digital
Digital (142, 166, 21847)-net over F4, using
- t-expansion [i] based on digital (141, 166, 21847)-net over F4, using
- net defined by OOA [i] based on linear OOA(4166, 21847, F4, 25, 25) (dual of [(21847, 25), 546009, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4166, 262165, F4, 25) (dual of [262165, 261999, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(4166, 262165, F4, 25) (dual of [262165, 261999, 26]-code), using
- net defined by OOA [i] based on linear OOA(4166, 21847, F4, 25, 25) (dual of [(21847, 25), 546009, 26]-NRT-code), using
(166−24, 166, 131083)-Net over F4 — Digital
Digital (142, 166, 131083)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4166, 131083, F4, 2, 24) (dual of [(131083, 2), 262000, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4166, 262166, F4, 24) (dual of [262166, 262000, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(4163, 262145, F4, 25) (dual of [262145, 261982, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(4145, 262145, F4, 21) (dual of [262145, 262000, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- OOA 2-folding [i] based on linear OA(4166, 262166, F4, 24) (dual of [262166, 262000, 25]-code), using
(166−24, 166, large)-Net in Base 4 — Upper bound on s
There is no (142, 166, large)-net in base 4, because
- 22 times m-reduction [i] would yield (142, 144, large)-net in base 4, but