Best Known (76−13, 76, s)-Nets in Base 4
(76−13, 76, 10925)-Net over F4 — Constructive and digital
Digital (63, 76, 10925)-net over F4, using
- net defined by OOA [i] based on linear OOA(476, 10925, F4, 13, 13) (dual of [(10925, 13), 141949, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(476, 65551, F4, 13) (dual of [65551, 65475, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(476, 65555, F4, 13) (dual of [65555, 65479, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(476, 65555, F4, 13) (dual of [65555, 65479, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(476, 65551, F4, 13) (dual of [65551, 65475, 14]-code), using
(76−13, 76, 32777)-Net over F4 — Digital
Digital (63, 76, 32777)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(476, 32777, F4, 2, 13) (dual of [(32777, 2), 65478, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(476, 65554, F4, 13) (dual of [65554, 65478, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(476, 65555, F4, 13) (dual of [65555, 65479, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(476, 65555, F4, 13) (dual of [65555, 65479, 14]-code), using
- OOA 2-folding [i] based on linear OA(476, 65554, F4, 13) (dual of [65554, 65478, 14]-code), using
(76−13, 76, large)-Net in Base 4 — Upper bound on s
There is no (63, 76, large)-net in base 4, because
- 11 times m-reduction [i] would yield (63, 65, large)-net in base 4, but