Best Known (164, 164+32, s)-Nets in Base 4
(164, 164+32, 4097)-Net over F4 — Constructive and digital
Digital (164, 196, 4097)-net over F4, using
- t-expansion [i] based on 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
- net defined by OOA [i] based on linear OOA(4196, 4097, F4, 33, 33) (dual of [(4097, 33), 135005, 34]-NRT-code), using
(164, 164+32, 32866)-Net over F4 — Digital
Digital (164, 196, 32866)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4196, 32866, F4, 32) (dual of [32866, 32670, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4196, 65556, F4, 32) (dual of [65556, 65360, 33]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(4193, 65537, F4, 33) (dual of [65537, 65344, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(4177, 65537, F4, 29) (dual of [65537, 65360, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [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 C([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4196, 65556, F4, 32) (dual of [65556, 65360, 33]-code), using
(164, 164+32, large)-Net in Base 4 — Upper bound on s
There is no (164, 196, large)-net in base 4, because
- 30 times m-reduction [i] would yield (164, 166, large)-net in base 4, but