Best Known (71−9, 71, s)-Nets in Base 4
(71−9, 71, 1048582)-Net over F4 — Constructive and digital
Digital (62, 71, 1048582)-net over F4, using
- net defined by OOA [i] based on linear OOA(471, 1048582, F4, 9, 9) (dual of [(1048582, 9), 9437167, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(471, 4194329, F4, 9) (dual of [4194329, 4194258, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(471, 4194330, F4, 9) (dual of [4194330, 4194259, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(44, 26, F4, 2) (dual of [26, 22, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- Hamming code H(4,4) [i]
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(471, 4194330, F4, 9) (dual of [4194330, 4194259, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(471, 4194329, F4, 9) (dual of [4194329, 4194258, 10]-code), using
(71−9, 71, 2097165)-Net over F4 — Digital
Digital (62, 71, 2097165)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(471, 2097165, F4, 2, 9) (dual of [(2097165, 2), 4194259, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(471, 4194330, F4, 9) (dual of [4194330, 4194259, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(467, 4194304, F4, 9) (dual of [4194304, 4194237, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(445, 4194304, F4, 6) (dual of [4194304, 4194259, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(44, 26, F4, 2) (dual of [26, 22, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- Hamming code H(4,4) [i]
- discarding factors / shortening the dual code based on linear OA(44, 85, F4, 2) (dual of [85, 81, 3]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- OOA 2-folding [i] based on linear OA(471, 4194330, F4, 9) (dual of [4194330, 4194259, 10]-code), using
(71−9, 71, large)-Net in Base 4 — Upper bound on s
There is no (62, 71, large)-net in base 4, because
- 7 times m-reduction [i] would yield (62, 64, large)-net in base 4, but