Best Known (171−28, 171, s)-Nets in Base 4
(171−28, 171, 4682)-Net over F4 — Constructive and digital
Digital (143, 171, 4682)-net over F4, using
- 1 times m-reduction [i] based on digital (143, 172, 4682)-net over F4, using
- net defined by OOA [i] based on linear OOA(4172, 4682, F4, 29, 29) (dual of [(4682, 29), 135606, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4172, 65549, F4, 29) (dual of [65549, 65377, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4172, 65555, F4, 29) (dual of [65555, 65383, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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, 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(28) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4172, 65555, F4, 29) (dual of [65555, 65383, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4172, 65549, F4, 29) (dual of [65549, 65377, 30]-code), using
- net defined by OOA [i] based on linear OOA(4172, 4682, F4, 29, 29) (dual of [(4682, 29), 135606, 30]-NRT-code), using
(171−28, 171, 32777)-Net over F4 — Digital
Digital (143, 171, 32777)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4171, 32777, F4, 2, 28) (dual of [(32777, 2), 65383, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4171, 65554, F4, 28) (dual of [65554, 65383, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4171, 65555, F4, 28) (dual of [65555, 65384, 29]-code), using
- construction XX applied to Ce(28) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(4145, 65536, F4, 25) (dual of [65536, 65391, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(28) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4171, 65555, F4, 28) (dual of [65555, 65384, 29]-code), using
- OOA 2-folding [i] based on linear OA(4171, 65554, F4, 28) (dual of [65554, 65383, 29]-code), using
(171−28, 171, large)-Net in Base 4 — Upper bound on s
There is no (143, 171, large)-net in base 4, because
- 26 times m-reduction [i] would yield (143, 145, large)-net in base 4, but