Best Known (65, 65+10, s)-Nets in Base 4
(65, 65+10, 209720)-Net over F4 — Constructive and digital
Digital (65, 75, 209720)-net over F4, using
- net defined by OOA [i] based on linear OOA(475, 209720, F4, 10, 10) (dual of [(209720, 10), 2097125, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(475, 1048600, F4, 10) (dual of [1048600, 1048525, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- OA 5-folding and stacking [i] based on linear OA(475, 1048600, F4, 10) (dual of [1048600, 1048525, 11]-code), using
(65, 65+10, 524300)-Net over F4 — Digital
Digital (65, 75, 524300)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(475, 524300, F4, 2, 10) (dual of [(524300, 2), 1048525, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(475, 1048600, F4, 10) (dual of [1048600, 1048525, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(451, 1048576, F4, 7) (dual of [1048576, 1048525, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(475, 1048600, F4, 10) (dual of [1048600, 1048525, 11]-code), using
(65, 65+10, large)-Net in Base 4 — Upper bound on s
There is no (65, 75, large)-net in base 4, because
- 8 times m-reduction [i] would yield (65, 67, large)-net in base 4, but