Best Known (216, 216+29, s)-Nets in Base 4
(216, 216+29, 299597)-Net over F4 — Constructive and digital
Digital (216, 245, 299597)-net over F4, using
- 42 times duplication [i] based on digital (214, 243, 299597)-net over F4, using
- net defined by OOA [i] based on linear OOA(4243, 299597, F4, 29, 29) (dual of [(299597, 29), 8688070, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4243, 4194359, F4, 29) (dual of [4194359, 4194116, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4242, 4194358, F4, 29) (dual of [4194358, 4194116, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4188, 4194304, F4, 23) (dual of [4194304, 4194116, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(410, 54, F4, 5) (dual of [54, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4242, 4194358, F4, 29) (dual of [4194358, 4194116, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4243, 4194359, F4, 29) (dual of [4194359, 4194116, 30]-code), using
- net defined by OOA [i] based on linear OOA(4243, 299597, F4, 29, 29) (dual of [(299597, 29), 8688070, 30]-NRT-code), using
(216, 216+29, 1489742)-Net over F4 — Digital
Digital (216, 245, 1489742)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4245, 1489742, F4, 2, 29) (dual of [(1489742, 2), 2979239, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4245, 2097186, F4, 2, 29) (dual of [(2097186, 2), 4194127, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4245, 4194372, F4, 29) (dual of [4194372, 4194127, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(4232, 4194304, F4, 29) (dual of [4194304, 4194072, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(413, 68, F4, 6) (dual of [68, 55, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 70, F4, 6) (dual of [70, 57, 7]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(4245, 4194372, F4, 29) (dual of [4194372, 4194127, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(4245, 2097186, F4, 2, 29) (dual of [(2097186, 2), 4194127, 30]-NRT-code), using
(216, 216+29, large)-Net in Base 4 — Upper bound on s
There is no (216, 245, large)-net in base 4, because
- 27 times m-reduction [i] would yield (216, 218, large)-net in base 4, but