Best Known (157−24, 157, s)-Nets in Base 3
(157−24, 157, 1647)-Net over F3 — Constructive and digital
Digital (133, 157, 1647)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (120, 144, 1640)-net over F3, using
- net defined by OOA [i] based on linear OOA(3144, 1640, F3, 24, 24) (dual of [(1640, 24), 39216, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3144, 19680, F3, 24) (dual of [19680, 19536, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 19683, F3, 24) (dual of [19683, 19539, 25]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 19683, F3, 24) (dual of [19683, 19539, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3144, 19680, F3, 24) (dual of [19680, 19536, 25]-code), using
- net defined by OOA [i] based on linear OOA(3144, 1640, F3, 24, 24) (dual of [(1640, 24), 39216, 25]-NRT-code), using
- digital (1, 13, 7)-net over F3, using
(157−24, 157, 10920)-Net over F3 — Digital
Digital (133, 157, 10920)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3157, 10920, F3, 24) (dual of [10920, 10763, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 19732, F3, 24) (dual of [19732, 19575, 25]-code), using
- strength reduction [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
- strength reduction [i] based on linear OA(3157, 19732, F3, 25) (dual of [19732, 19575, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3157, 19732, F3, 24) (dual of [19732, 19575, 25]-code), using
(157−24, 157, 4620263)-Net in Base 3 — Upper bound on s
There is no (133, 157, 4620264)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 809 165682 504742 721349 095972 097765 637599 337044 981511 410123 285282 536671 455809 > 3157 [i]