Best Known (124−21, 124, s)-Nets in Base 4
(124−21, 124, 6555)-Net over F4 — Constructive and digital
Digital (103, 124, 6555)-net over F4, using
- net defined by OOA [i] based on linear OOA(4124, 6555, F4, 21, 21) (dual of [(6555, 21), 137531, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4124, 65551, F4, 21) (dual of [65551, 65427, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4124, 65555, F4, 21) (dual of [65555, 65431, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4124, 65555, F4, 21) (dual of [65555, 65431, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4124, 65551, F4, 21) (dual of [65551, 65427, 22]-code), using
(124−21, 124, 30300)-Net over F4 — Digital
Digital (103, 124, 30300)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4124, 30300, F4, 2, 21) (dual of [(30300, 2), 60476, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4124, 32777, F4, 2, 21) (dual of [(32777, 2), 65430, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4124, 65554, F4, 21) (dual of [65554, 65430, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4124, 65555, F4, 21) (dual of [65555, 65431, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(20) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4124, 65555, F4, 21) (dual of [65555, 65431, 22]-code), using
- OOA 2-folding [i] based on linear OA(4124, 65554, F4, 21) (dual of [65554, 65430, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(4124, 32777, F4, 2, 21) (dual of [(32777, 2), 65430, 22]-NRT-code), using
(124−21, 124, large)-Net in Base 4 — Upper bound on s
There is no (103, 124, large)-net in base 4, because
- 19 times m-reduction [i] would yield (103, 105, large)-net in base 4, but