Best Known (70, 70+14, s)-Nets in Base 4
(70, 70+14, 9365)-Net over F4 — Constructive and digital
Digital (70, 84, 9365)-net over F4, using
- net defined by OOA [i] based on linear OOA(484, 9365, F4, 14, 14) (dual of [(9365, 14), 131026, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(484, 65555, F4, 14) (dual of [65555, 65471, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(465, 65536, F4, 11) (dual of [65536, 65471, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- OA 7-folding and stacking [i] based on linear OA(484, 65555, F4, 14) (dual of [65555, 65471, 15]-code), using
(70, 70+14, 32777)-Net over F4 — Digital
Digital (70, 84, 32777)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(484, 32777, F4, 2, 14) (dual of [(32777, 2), 65470, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(484, 65554, F4, 14) (dual of [65554, 65470, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(484, 65555, F4, 14) (dual of [65555, 65471, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(465, 65536, F4, 11) (dual of [65536, 65471, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(484, 65555, F4, 14) (dual of [65555, 65471, 15]-code), using
- OOA 2-folding [i] based on linear OA(484, 65554, F4, 14) (dual of [65554, 65470, 15]-code), using
(70, 70+14, large)-Net in Base 4 — Upper bound on s
There is no (70, 84, large)-net in base 4, because
- 12 times m-reduction [i] would yield (70, 72, large)-net in base 4, but