Best Known (167−25, 167, s)-Nets in Base 4
(167−25, 167, 21847)-Net over F4 — Constructive and digital
Digital (142, 167, 21847)-net over F4, using
- 41 times duplication [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
(167−25, 167, 98879)-Net over F4 — Digital
Digital (142, 167, 98879)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4167, 98879, F4, 2, 25) (dual of [(98879, 2), 197591, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4167, 131083, F4, 2, 25) (dual of [(131083, 2), 261999, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4167, 262166, F4, 25) (dual of [262166, 261999, 26]-code), using
- 1 times code embedding in larger space [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
- 1 times code embedding in larger space [i] based on linear OA(4166, 262165, F4, 25) (dual of [262165, 261999, 26]-code), using
- OOA 2-folding [i] based on linear OA(4167, 262166, F4, 25) (dual of [262166, 261999, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(4167, 131083, F4, 2, 25) (dual of [(131083, 2), 261999, 26]-NRT-code), using
(167−25, 167, large)-Net in Base 4 — Upper bound on s
There is no (142, 167, large)-net in base 4, because
- 23 times m-reduction [i] would yield (142, 144, large)-net in base 4, but