Best Known (159−25, 159, s)-Nets in Base 3
(159−25, 159, 1648)-Net over F3 — Constructive and digital
Digital (134, 159, 1648)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (120, 145, 1640)-net over F3, using
- net defined by OOA [i] based on linear OOA(3145, 1640, F3, 25, 25) (dual of [(1640, 25), 40855, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3145, 19681, F3, 25) (dual of [19681, 19536, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3145, 19681, F3, 25) (dual of [19681, 19536, 26]-code), using
- net defined by OOA [i] based on linear OOA(3145, 1640, F3, 25, 25) (dual of [(1640, 25), 40855, 26]-NRT-code), using
- digital (2, 14, 8)-net over F3, using
(159−25, 159, 9867)-Net over F3 — Digital
Digital (134, 159, 9867)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3159, 9867, F3, 2, 25) (dual of [(9867, 2), 19575, 26]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3157, 9866, F3, 2, 25) (dual of [(9866, 2), 19575, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3157, 19732, F3, 25) (dual of [19732, 19575, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3109, 19684, F3, 19) (dual of [19684, 19575, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(312, 48, F3, 5) (dual of [48, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- OOA 2-folding [i] based on linear OA(3157, 19732, F3, 25) (dual of [19732, 19575, 26]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3157, 9866, F3, 2, 25) (dual of [(9866, 2), 19575, 26]-NRT-code), using
(159−25, 159, 5063221)-Net in Base 3 — Upper bound on s
There is no (134, 159, 5063222)-net in base 3, because
- 1 times m-reduction [i] would yield (134, 158, 5063222)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2427 495632 397378 640665 794777 548905 792827 268306 093382 249528 338554 829754 542681 > 3158 [i]