Best Known (165, 165+27, s)-Nets in Base 4
(165, 165+27, 20169)-Net over F4 — Constructive and digital
Digital (165, 192, 20169)-net over F4, using
- 41 times duplication [i] based on digital (164, 191, 20169)-net over F4, using
- net defined by OOA [i] based on linear OOA(4191, 20169, F4, 27, 27) (dual of [(20169, 27), 544372, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4191, 262198, F4, 27) (dual of [262198, 262007, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 262199, F4, 27) (dual of [262199, 262008, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- 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(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 262199, F4, 27) (dual of [262199, 262008, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4191, 262198, F4, 27) (dual of [262198, 262007, 28]-code), using
- net defined by OOA [i] based on linear OOA(4191, 20169, F4, 27, 27) (dual of [(20169, 27), 544372, 28]-NRT-code), using
(165, 165+27, 134952)-Net over F4 — Digital
Digital (165, 192, 134952)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4192, 134952, F4, 27) (dual of [134952, 134760, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4192, 262200, F4, 27) (dual of [262200, 262008, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4191, 262199, F4, 27) (dual of [262199, 262008, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- 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(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4191, 262199, F4, 27) (dual of [262199, 262008, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4192, 262200, F4, 27) (dual of [262200, 262008, 28]-code), using
(165, 165+27, large)-Net in Base 4 — Upper bound on s
There is no (165, 192, large)-net in base 4, because
- 25 times m-reduction [i] would yield (165, 167, large)-net in base 4, but