Best Known (163, 163+28, s)-Nets in Base 4
(163, 163+28, 18725)-Net over F4 — Constructive and digital
Digital (163, 191, 18725)-net over F4, using
- t-expansion [i] based on digital (162, 191, 18725)-net over F4, using
- net defined by OOA [i] based on linear OOA(4191, 18725, F4, 29, 29) (dual of [(18725, 29), 542834, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4191, 262151, F4, 29) (dual of [262151, 261960, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 262154, F4, 29) (dual of [262154, 261963, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4181, 262144, F4, 27) (dual of [262144, 261963, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 262154, F4, 29) (dual of [262154, 261963, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4191, 262151, F4, 29) (dual of [262151, 261960, 30]-code), using
- net defined by OOA [i] based on linear OOA(4191, 18725, F4, 29, 29) (dual of [(18725, 29), 542834, 30]-NRT-code), using
(163, 163+28, 120783)-Net over F4 — Digital
Digital (163, 191, 120783)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4191, 120783, F4, 2, 28) (dual of [(120783, 2), 241375, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4191, 131081, F4, 2, 28) (dual of [(131081, 2), 261971, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4191, 262162, F4, 28) (dual of [262162, 261971, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 262163, F4, 28) (dual of [262163, 261972, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(41, 19, F4, 1) (dual of [19, 18, 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(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 262163, F4, 28) (dual of [262163, 261972, 29]-code), using
- OOA 2-folding [i] based on linear OA(4191, 262162, F4, 28) (dual of [262162, 261971, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(4191, 131081, F4, 2, 28) (dual of [(131081, 2), 261971, 29]-NRT-code), using
(163, 163+28, large)-Net in Base 4 — Upper bound on s
There is no (163, 191, large)-net in base 4, because
- 26 times m-reduction [i] would yield (163, 165, large)-net in base 4, but