Best Known (221, 221+34, s)-Nets in Base 4
(221, 221+34, 61682)-Net over F4 — Constructive and digital
Digital (221, 255, 61682)-net over F4, using
- 41 times duplication [i] based on digital (220, 254, 61682)-net over F4, using
- net defined by OOA [i] based on linear OOA(4254, 61682, F4, 34, 34) (dual of [(61682, 34), 2096934, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(4254, 1048594, F4, 34) (dual of [1048594, 1048340, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4254, 1048597, F4, 34) (dual of [1048597, 1048343, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- linear OA(4251, 1048576, F4, 34) (dual of [1048576, 1048325, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(33) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4254, 1048597, F4, 34) (dual of [1048597, 1048343, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(4254, 1048594, F4, 34) (dual of [1048594, 1048340, 35]-code), using
- net defined by OOA [i] based on linear OOA(4254, 61682, F4, 34, 34) (dual of [(61682, 34), 2096934, 35]-NRT-code), using
(221, 221+34, 349533)-Net over F4 — Digital
Digital (221, 255, 349533)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4255, 349533, F4, 3, 34) (dual of [(349533, 3), 1048344, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4255, 1048599, F4, 34) (dual of [1048599, 1048344, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4255, 1048600, F4, 34) (dual of [1048600, 1048345, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- linear OA(4251, 1048576, F4, 34) (dual of [1048576, 1048325, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(44, 24, F4, 2) (dual of [24, 20, 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(33) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(4255, 1048600, F4, 34) (dual of [1048600, 1048345, 35]-code), using
- OOA 3-folding [i] based on linear OA(4255, 1048599, F4, 34) (dual of [1048599, 1048344, 35]-code), using
(221, 221+34, large)-Net in Base 4 — Upper bound on s
There is no (221, 255, large)-net in base 4, because
- 32 times m-reduction [i] would yield (221, 223, large)-net in base 4, but