Best Known (50, 50+9, s)-Nets in Base 4
(50, 50+9, 65541)-Net over F4 — Constructive and digital
Digital (50, 59, 65541)-net over F4, using
- 41 times duplication [i] based on digital (49, 58, 65541)-net over F4, using
- net defined by OOA [i] based on linear OOA(458, 65541, F4, 9, 9) (dual of [(65541, 9), 589811, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(458, 262165, F4, 9) (dual of [262165, 262107, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(455, 262144, F4, 9) (dual of [262144, 262089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(8) ⊂ Ce(5) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(458, 262165, F4, 9) (dual of [262165, 262107, 10]-code), using
- net defined by OOA [i] based on linear OOA(458, 65541, F4, 9, 9) (dual of [(65541, 9), 589811, 10]-NRT-code), using
(50, 50+9, 131083)-Net over F4 — Digital
Digital (50, 59, 131083)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(459, 131083, F4, 2, 9) (dual of [(131083, 2), 262107, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(459, 262166, F4, 9) (dual of [262166, 262107, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(458, 262165, F4, 9) (dual of [262165, 262107, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(455, 262144, F4, 9) (dual of [262144, 262089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(8) ⊂ Ce(5) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(458, 262165, F4, 9) (dual of [262165, 262107, 10]-code), using
- OOA 2-folding [i] based on linear OA(459, 262166, F4, 9) (dual of [262166, 262107, 10]-code), using
(50, 50+9, large)-Net in Base 4 — Upper bound on s
There is no (50, 59, large)-net in base 4, because
- 7 times m-reduction [i] would yield (50, 52, large)-net in base 4, but