Best Known (70−9, 70, s)-Nets in Base 4
(70−9, 70, 1048581)-Net over F4 — Constructive and digital
Digital (61, 70, 1048581)-net over F4, using
- net defined by OOA [i] based on linear OOA(470, 1048581, F4, 9, 9) (dual of [(1048581, 9), 9437159, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(470, 4194325, F4, 9) (dual of [4194325, 4194255, 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(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(470, 4194325, F4, 9) (dual of [4194325, 4194255, 10]-code), using
(70−9, 70, 2097162)-Net over F4 — Digital
Digital (61, 70, 2097162)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(470, 2097162, F4, 2, 9) (dual of [(2097162, 2), 4194254, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(470, 4194324, F4, 9) (dual of [4194324, 4194254, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(470, 4194325, F4, 9) (dual of [4194325, 4194255, 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(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
- discarding factors / shortening the dual code based on linear OA(470, 4194325, F4, 9) (dual of [4194325, 4194255, 10]-code), using
- OOA 2-folding [i] based on linear OA(470, 4194324, F4, 9) (dual of [4194324, 4194254, 10]-code), using
(70−9, 70, large)-Net in Base 4 — Upper bound on s
There is no (61, 70, large)-net in base 4, because
- 7 times m-reduction [i] would yield (61, 63, large)-net in base 4, but