Best Known (131, 156, s)-Nets in Base 2
(131, 156, 480)-Net over F2 — Constructive and digital
Digital (131, 156, 480)-net over F2, using
- trace code for nets [i] based on digital (1, 26, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
(131, 156, 1324)-Net over F2 — Digital
Digital (131, 156, 1324)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2156, 1324, F2, 3, 25) (dual of [(1324, 3), 3816, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2156, 1377, F2, 3, 25) (dual of [(1377, 3), 3975, 26]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2155, 1377, F2, 3, 25) (dual of [(1377, 3), 3976, 26]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2152, 1376, F2, 3, 25) (dual of [(1376, 3), 3976, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2152, 4128, F2, 25) (dual of [4128, 3976, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2151, 4127, F2, 25) (dual of [4127, 3976, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(2145, 4097, F2, 25) (dual of [4097, 3952, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(2121, 4097, F2, 21) (dual of [4097, 3976, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2151, 4127, F2, 25) (dual of [4127, 3976, 26]-code), using
- OOA 3-folding [i] based on linear OA(2152, 4128, F2, 25) (dual of [4128, 3976, 26]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2152, 1376, F2, 3, 25) (dual of [(1376, 3), 3976, 26]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2155, 1377, F2, 3, 25) (dual of [(1377, 3), 3976, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2156, 1377, F2, 3, 25) (dual of [(1377, 3), 3975, 26]-NRT-code), using
(131, 156, 40877)-Net in Base 2 — Upper bound on s
There is no (131, 156, 40878)-net in base 2, because
- 1 times m-reduction [i] would yield (131, 155, 40878)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 45684 499836 573112 507898 444613 186914 698532 195857 > 2155 [i]