Best Known (103, 123, s)-Nets in Base 4
(103, 123, 6555)-Net over F4 — Constructive and digital
Digital (103, 123, 6555)-net over F4, using
- 1 times m-reduction [i] based on 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
- net defined by OOA [i] based on linear OOA(4124, 6555, F4, 21, 21) (dual of [(6555, 21), 137531, 22]-NRT-code), using
(103, 123, 32777)-Net over F4 — Digital
Digital (103, 123, 32777)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4123, 32777, F4, 2, 20) (dual of [(32777, 2), 65431, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4123, 65554, F4, 20) (dual of [65554, 65431, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4123, 65555, F4, 20) (dual of [65555, 65432, 21]-code), using
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ Ce(16) [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(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(41, 18, F4, 1) (dual of [18, 17, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4123, 65555, F4, 20) (dual of [65555, 65432, 21]-code), using
- OOA 2-folding [i] based on linear OA(4123, 65554, F4, 20) (dual of [65554, 65431, 21]-code), using
(103, 123, large)-Net in Base 4 — Upper bound on s
There is no (103, 123, large)-net in base 4, because
- 18 times m-reduction [i] would yield (103, 105, large)-net in base 4, but