Best Known (50, 60, s)-Nets in Base 4
(50, 60, 13111)-Net over F4 — Constructive and digital
Digital (50, 60, 13111)-net over F4, using
- net defined by OOA [i] based on linear OOA(460, 13111, F4, 10, 10) (dual of [(13111, 10), 131050, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(460, 65555, F4, 10) (dual of [65555, 65495, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- OA 5-folding and stacking [i] based on linear OA(460, 65555, F4, 10) (dual of [65555, 65495, 11]-code), using
(50, 60, 34569)-Net over F4 — Digital
Digital (50, 60, 34569)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(460, 34569, F4, 10) (dual of [34569, 34509, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(460, 65555, F4, 10) (dual of [65555, 65495, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(43, 19, F4, 2) (dual of [19, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(460, 65555, F4, 10) (dual of [65555, 65495, 11]-code), using
(50, 60, large)-Net in Base 4 — Upper bound on s
There is no (50, 60, large)-net in base 4, because
- 8 times m-reduction [i] would yield (50, 52, large)-net in base 4, but