Best Known (141, 141+29, s)-Nets in Base 4
(141, 141+29, 4681)-Net over F4 — Constructive and digital
Digital (141, 170, 4681)-net over F4, using
- 41 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
(141, 141+29, 27295)-Net over F4 — Digital
Digital (141, 170, 27295)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4170, 27295, F4, 2, 29) (dual of [(27295, 2), 54420, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4170, 32772, F4, 2, 29) (dual of [(32772, 2), 65374, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4170, 65544, F4, 29) (dual of [65544, 65374, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4170, 65545, F4, 29) (dual of [65545, 65375, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [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(41, 9, F4, 1) (dual of [9, 8, 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
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4170, 65545, F4, 29) (dual of [65545, 65375, 30]-code), using
- OOA 2-folding [i] based on linear OA(4170, 65544, F4, 29) (dual of [65544, 65374, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(4170, 32772, F4, 2, 29) (dual of [(32772, 2), 65374, 30]-NRT-code), using
(141, 141+29, large)-Net in Base 4 — Upper bound on s
There is no (141, 170, large)-net in base 4, because
- 27 times m-reduction [i] would yield (141, 143, large)-net in base 4, but