Best Known (181−30, 181, s)-Nets in Base 4
(181−30, 181, 4370)-Net over F4 — Constructive and digital
Digital (151, 181, 4370)-net over F4, using
- 41 times duplication [i] based on digital (150, 180, 4370)-net over F4, using
- net defined by OOA [i] based on linear OOA(4180, 4370, F4, 30, 30) (dual of [(4370, 30), 130920, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4180, 65550, F4, 30) (dual of [65550, 65370, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4180, 65555, F4, 30) (dual of [65555, 65375, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(26) [i] based on
- linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(29) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4180, 65555, F4, 30) (dual of [65555, 65375, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(4180, 65550, F4, 30) (dual of [65550, 65370, 31]-code), using
- net defined by OOA [i] based on linear OOA(4180, 4370, F4, 30, 30) (dual of [(4370, 30), 130920, 31]-NRT-code), using
(181−30, 181, 32778)-Net over F4 — Digital
Digital (151, 181, 32778)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4181, 32778, F4, 2, 30) (dual of [(32778, 2), 65375, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4181, 65556, F4, 30) (dual of [65556, 65375, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 65557, F4, 30) (dual of [65557, 65376, 31]-code), using
- construction XX applied to Ce(29) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4161, 65536, F4, 27) (dual of [65536, 65375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4153, 65536, F4, 26) (dual of [65536, 65383, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(29) ⊂ Ce(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4181, 65557, F4, 30) (dual of [65557, 65376, 31]-code), using
- OOA 2-folding [i] based on linear OA(4181, 65556, F4, 30) (dual of [65556, 65375, 31]-code), using
(181−30, 181, large)-Net in Base 4 — Upper bound on s
There is no (151, 181, large)-net in base 4, because
- 28 times m-reduction [i] would yield (151, 153, large)-net in base 4, but