Best Known (218−28, 218, s)-Nets in Base 4
(218−28, 218, 74901)-Net over F4 — Constructive and digital
Digital (190, 218, 74901)-net over F4, using
- net defined by OOA [i] based on linear OOA(4218, 74901, F4, 28, 28) (dual of [(74901, 28), 2097010, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(4218, 1048614, F4, 28) (dual of [1048614, 1048396, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4218, 1048619, F4, 28) (dual of [1048619, 1048401, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4218, 1048619, F4, 28) (dual of [1048619, 1048401, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(4218, 1048614, F4, 28) (dual of [1048614, 1048396, 29]-code), using
(218−28, 218, 524309)-Net over F4 — Digital
Digital (190, 218, 524309)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4218, 524309, F4, 2, 28) (dual of [(524309, 2), 1048400, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4218, 1048618, F4, 28) (dual of [1048618, 1048400, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4218, 1048619, F4, 28) (dual of [1048619, 1048401, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4218, 1048619, F4, 28) (dual of [1048619, 1048401, 29]-code), using
- OOA 2-folding [i] based on linear OA(4218, 1048618, F4, 28) (dual of [1048618, 1048400, 29]-code), using
(218−28, 218, large)-Net in Base 4 — Upper bound on s
There is no (190, 218, large)-net in base 4, because
- 26 times m-reduction [i] would yield (190, 192, large)-net in base 4, but