Best Known (168, 168+26, s)-Nets in Base 4
(168, 168+26, 80661)-Net over F4 — Constructive and digital
Digital (168, 194, 80661)-net over F4, using
- net defined by OOA [i] based on linear OOA(4194, 80661, F4, 26, 26) (dual of [(80661, 26), 2096992, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4194, 1048593, F4, 26) (dual of [1048593, 1048399, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4194, 1048597, F4, 26) (dual of [1048597, 1048403, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4194, 1048597, F4, 26) (dual of [1048597, 1048403, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4194, 1048593, F4, 26) (dual of [1048593, 1048399, 27]-code), using
(168, 168+26, 349532)-Net over F4 — Digital
Digital (168, 194, 349532)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4194, 349532, F4, 3, 26) (dual of [(349532, 3), 1048402, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4194, 1048596, F4, 26) (dual of [1048596, 1048402, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4194, 1048597, F4, 26) (dual of [1048597, 1048403, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4194, 1048597, F4, 26) (dual of [1048597, 1048403, 27]-code), using
- OOA 3-folding [i] based on linear OA(4194, 1048596, F4, 26) (dual of [1048596, 1048402, 27]-code), using
(168, 168+26, large)-Net in Base 4 — Upper bound on s
There is no (168, 194, large)-net in base 4, because
- 24 times m-reduction [i] would yield (168, 170, large)-net in base 4, but