Best Known (155−21, 155, s)-Nets in Base 4
(155−21, 155, 104859)-Net over F4 — Constructive and digital
Digital (134, 155, 104859)-net over F4, using
- 41 times duplication [i] based on digital (133, 154, 104859)-net over F4, using
- net defined by OOA [i] based on linear OOA(4154, 104859, F4, 21, 21) (dual of [(104859, 21), 2201885, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4154, 1048591, F4, 21) (dual of [1048591, 1048437, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4154, 1048597, F4, 21) (dual of [1048597, 1048443, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4154, 1048597, F4, 21) (dual of [1048597, 1048443, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4154, 1048591, F4, 21) (dual of [1048591, 1048437, 22]-code), using
- net defined by OOA [i] based on linear OOA(4154, 104859, F4, 21, 21) (dual of [(104859, 21), 2201885, 22]-NRT-code), using
(155−21, 155, 349533)-Net over F4 — Digital
Digital (134, 155, 349533)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4155, 349533, F4, 3, 21) (dual of [(349533, 3), 1048444, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4155, 1048599, F4, 21) (dual of [1048599, 1048444, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4155, 1048600, F4, 21) (dual of [1048600, 1048445, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4155, 1048600, F4, 21) (dual of [1048600, 1048445, 22]-code), using
- OOA 3-folding [i] based on linear OA(4155, 1048599, F4, 21) (dual of [1048599, 1048444, 22]-code), using
(155−21, 155, large)-Net in Base 4 — Upper bound on s
There is no (134, 155, large)-net in base 4, because
- 19 times m-reduction [i] would yield (134, 136, large)-net in base 4, but