Best Known (51, 51+10, s)-Nets in Base 4
(51, 51+10, 13111)-Net over F4 — Constructive and digital
Digital (51, 61, 13111)-net over F4, using
- 41 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(460, 13111, F4, 10, 10) (dual of [(13111, 10), 131050, 11]-NRT-code), using
(51, 51+10, 41111)-Net over F4 — Digital
Digital (51, 61, 41111)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(461, 41111, F4, 10) (dual of [41111, 41050, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(461, 65557, F4, 10) (dual of [65557, 65496, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(6) ⊂ Ce(5) [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(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(43, 20, F4, 2) (dual of [20, 17, 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
- 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(9) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(461, 65557, F4, 10) (dual of [65557, 65496, 11]-code), using
(51, 51+10, large)-Net in Base 4 — Upper bound on s
There is no (51, 61, large)-net in base 4, because
- 8 times m-reduction [i] would yield (51, 53, large)-net in base 4, but