Best Known (161−24, 161, s)-Nets in Base 4
(161−24, 161, 5478)-Net over F4 — Constructive and digital
Digital (137, 161, 5478)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (5, 17, 17)-net over F4, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 5 and N(F) ≥ 17, using
- net from sequence [i] based on digital (5, 16)-sequence over F4, using
- digital (120, 144, 5461)-net over F4, using
- net defined by OOA [i] based on linear OOA(4144, 5461, F4, 24, 24) (dual of [(5461, 24), 130920, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4144, 65532, F4, 24) (dual of [65532, 65388, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4144, 65536, F4, 24) (dual of [65536, 65392, 25]-code), using
- 1 times truncation [i] based on linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4144, 65536, F4, 24) (dual of [65536, 65392, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4144, 65532, F4, 24) (dual of [65532, 65388, 25]-code), using
- net defined by OOA [i] based on linear OOA(4144, 5461, F4, 24, 24) (dual of [(5461, 24), 130920, 25]-NRT-code), using
- digital (5, 17, 17)-net over F4, using
(161−24, 161, 65603)-Net over F4 — Digital
Digital (137, 161, 65603)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4161, 65603, F4, 24) (dual of [65603, 65442, 25]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4158, 65598, F4, 24) (dual of [65598, 65440, 25]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(4145, 65537, F4, 25) (dual of [65537, 65392, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(497, 65537, F4, 17) (dual of [65537, 65440, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(413, 61, F4, 6) (dual of [61, 48, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to C([0,12]) ⊂ C([0,8]) [i] based on
- linear OA(4158, 65600, F4, 22) (dual of [65600, 65442, 23]-code), using Gilbert–Varšamov bound and bm = 4158 > Vbs−1(k−1) = 29 154121 380868 416756 039450 918797 313669 619952 178752 233686 921655 713323 373598 562591 077668 644578 [i]
- linear OA(41, 3, F4, 1) (dual of [3, 2, 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
- linear OA(4158, 65598, F4, 24) (dual of [65598, 65440, 25]-code), using
- construction X with Varšamov bound [i] based on
(161−24, 161, large)-Net in Base 4 — Upper bound on s
There is no (137, 161, large)-net in base 4, because
- 22 times m-reduction [i] would yield (137, 139, large)-net in base 4, but