Best Known (163, 163+33, s)-Nets in Base 4
(163, 163+33, 4097)-Net over F4 — Constructive and digital
Digital (163, 196, 4097)-net over F4, using
- net defined by OOA [i] based on linear OOA(4196, 4097, F4, 33, 33) (dual of [(4097, 33), 135005, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4196, 65553, F4, 33) (dual of [65553, 65357, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4196, 65555, F4, 33) (dual of [65555, 65359, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 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(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(32) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4196, 65555, F4, 33) (dual of [65555, 65359, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(4196, 65553, F4, 33) (dual of [65553, 65357, 34]-code), using
(163, 163+33, 31380)-Net over F4 — Digital
Digital (163, 196, 31380)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4196, 31380, F4, 2, 33) (dual of [(31380, 2), 62564, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4196, 32777, F4, 2, 33) (dual of [(32777, 2), 65358, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4196, 65554, F4, 33) (dual of [65554, 65358, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4196, 65555, F4, 33) (dual of [65555, 65359, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 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(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(32) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4196, 65555, F4, 33) (dual of [65555, 65359, 34]-code), using
- OOA 2-folding [i] based on linear OA(4196, 65554, F4, 33) (dual of [65554, 65358, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(4196, 32777, F4, 2, 33) (dual of [(32777, 2), 65358, 34]-NRT-code), using
(163, 163+33, large)-Net in Base 4 — Upper bound on s
There is no (163, 196, large)-net in base 4, because
- 31 times m-reduction [i] would yield (163, 165, large)-net in base 4, but