Best Known (216, 216+30, s)-Nets in Base 4
(216, 216+30, 279621)-Net over F4 — Constructive and digital
Digital (216, 246, 279621)-net over F4, using
- 43 times duplication [i] based on digital (213, 243, 279621)-net over F4, using
- net defined by OOA [i] based on linear OOA(4243, 279621, F4, 30, 30) (dual of [(279621, 30), 8388387, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4243, 4194315, F4, 30) (dual of [4194315, 4194072, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- 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(40, 11, F4, 0) (dual of [11, 11, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- OA 15-folding and stacking [i] based on linear OA(4243, 4194315, F4, 30) (dual of [4194315, 4194072, 31]-code), using
- net defined by OOA [i] based on linear OOA(4243, 279621, F4, 30, 30) (dual of [(279621, 30), 8388387, 31]-NRT-code), using
(216, 216+30, 1398108)-Net over F4 — Digital
Digital (216, 246, 1398108)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4246, 1398108, F4, 3, 30) (dual of [(1398108, 3), 4194078, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4246, 4194324, F4, 30) (dual of [4194324, 4194078, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4246, 4194325, F4, 30) (dual of [4194325, 4194079, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- linear OA(4243, 4194304, F4, 30) (dual of [4194304, 4194061, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4221, 4194304, F4, 27) (dual of [4194304, 4194083, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(29) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4246, 4194325, F4, 30) (dual of [4194325, 4194079, 31]-code), using
- OOA 3-folding [i] based on linear OA(4246, 4194324, F4, 30) (dual of [4194324, 4194078, 31]-code), using
(216, 216+30, large)-Net in Base 4 — Upper bound on s
There is no (216, 246, large)-net in base 4, because
- 28 times m-reduction [i] would yield (216, 218, large)-net in base 4, but