Best Known (171−29, 171, s)-Nets in Base 4
(171−29, 171, 4681)-Net over F4 — Constructive and digital
Digital (142, 171, 4681)-net over F4, using
- 42 times duplication [i] based on digital (140, 169, 4681)-net over F4, using
- net defined by OOA [i] based on linear OOA(4169, 4681, F4, 29, 29) (dual of [(4681, 29), 135580, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4169, 65535, F4, 29) (dual of [65535, 65366, 30]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(4169, 65536, F4, 29) (dual of [65536, 65367, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4169, 65535, F4, 29) (dual of [65535, 65366, 30]-code), using
- net defined by OOA [i] based on linear OOA(4169, 4681, F4, 29, 29) (dual of [(4681, 29), 135580, 30]-NRT-code), using
(171−29, 171, 28791)-Net over F4 — Digital
Digital (142, 171, 28791)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4171, 28791, F4, 2, 29) (dual of [(28791, 2), 57411, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4171, 32773, F4, 2, 29) (dual of [(32773, 2), 65375, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4171, 65546, F4, 29) (dual of [65546, 65375, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4171, 65547, F4, 29) (dual of [65547, 65376, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(26) ⊂ 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(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(41, 10, F4, 1) (dual of [10, 9, 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(26) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4171, 65547, F4, 29) (dual of [65547, 65376, 30]-code), using
- OOA 2-folding [i] based on linear OA(4171, 65546, F4, 29) (dual of [65546, 65375, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(4171, 32773, F4, 2, 29) (dual of [(32773, 2), 65375, 30]-NRT-code), using
(171−29, 171, large)-Net in Base 4 — Upper bound on s
There is no (142, 171, large)-net in base 4, because
- 27 times m-reduction [i] would yield (142, 144, large)-net in base 4, but