Best Known (165, 165+33, s)-Nets in Base 4
(165, 165+33, 4097)-Net over F4 — Constructive and digital
Digital (165, 198, 4097)-net over F4, using
- 42 times duplication [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
(165, 165+33, 32782)-Net over F4 — Digital
Digital (165, 198, 32782)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4198, 32782, F4, 2, 33) (dual of [(32782, 2), 65366, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4198, 65564, F4, 33) (dual of [65564, 65366, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4198, 65565, F4, 33) (dual of [65565, 65367, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(28) [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(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(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(32) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4198, 65565, F4, 33) (dual of [65565, 65367, 34]-code), using
- OOA 2-folding [i] based on linear OA(4198, 65564, F4, 33) (dual of [65564, 65366, 34]-code), using
(165, 165+33, large)-Net in Base 4 — Upper bound on s
There is no (165, 198, large)-net in base 4, because
- 31 times m-reduction [i] would yield (165, 167, large)-net in base 4, but