Best Known (219, 219+41, s)-Nets in Base 2
(219, 219+41, 600)-Net over F2 — Constructive and digital
Digital (219, 260, 600)-net over F2, using
- trace code for nets [i] based on digital (11, 52, 120)-net over F32, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 11 and N(F) ≥ 120, using
- net from sequence [i] based on digital (11, 119)-sequence over F32, using
(219, 219+41, 1607)-Net over F2 — Digital
Digital (219, 260, 1607)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2260, 1607, F2, 2, 41) (dual of [(1607, 2), 2954, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2260, 2081, F2, 2, 41) (dual of [(2081, 2), 3902, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2260, 4162, F2, 41) (dual of [4162, 3902, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- linear OA(2241, 4097, F2, 41) (dual of [4097, 3856, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(2193, 4097, F2, 33) (dual of [4097, 3904, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(219, 65, F2, 7) (dual of [65, 46, 8]-code), using
- adding a parity check bit [i] based on linear OA(218, 64, F2, 6) (dual of [64, 46, 7]-code), using
- extracting embedded orthogonal array [i] based on digital (12, 18, 64)-net over F2, using
- adding a parity check bit [i] based on linear OA(218, 64, F2, 6) (dual of [64, 46, 7]-code), using
- construction X applied to C([0,20]) ⊂ C([0,16]) [i] based on
- OOA 2-folding [i] based on linear OA(2260, 4162, F2, 41) (dual of [4162, 3902, 42]-code), using
- discarding factors / shortening the dual code based on linear OOA(2260, 2081, F2, 2, 41) (dual of [(2081, 2), 3902, 42]-NRT-code), using
(219, 219+41, 65682)-Net in Base 2 — Upper bound on s
There is no (219, 260, 65683)-net in base 2, because
- 1 times m-reduction [i] would yield (219, 259, 65683)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 926476 453285 481582 919813 940921 392374 858169 412946 809051 983509 001728 859514 394728 > 2259 [i]