Best Known (256−42, 256, s)-Nets in Base 2
(256−42, 256, 520)-Net over F2 — Constructive and digital
Digital (214, 256, 520)-net over F2, using
- 21 times duplication [i] based on digital (213, 255, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 51, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- trace code for nets [i] based on digital (9, 51, 104)-net over F32, using
(256−42, 256, 1370)-Net over F2 — Digital
Digital (214, 256, 1370)-net over F2, using
- 21 times duplication [i] based on digital (213, 255, 1370)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2255, 1370, F2, 3, 42) (dual of [(1370, 3), 3855, 43]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2255, 4110, F2, 42) (dual of [4110, 3855, 43]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2253, 4108, F2, 42) (dual of [4108, 3855, 43]-code), using
- 1 times truncation [i] based on linear OA(2254, 4109, F2, 43) (dual of [4109, 3855, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(40) [i] based on
- linear OA(2253, 4096, F2, 43) (dual of [4096, 3843, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2241, 4096, F2, 41) (dual of [4096, 3855, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(42) ⊂ Ce(40) [i] based on
- 1 times truncation [i] based on linear OA(2254, 4109, F2, 43) (dual of [4109, 3855, 44]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2253, 4108, F2, 42) (dual of [4108, 3855, 43]-code), using
- OOA 3-folding [i] based on linear OA(2255, 4110, F2, 42) (dual of [4110, 3855, 43]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2255, 1370, F2, 3, 42) (dual of [(1370, 3), 3855, 43]-NRT-code), using
(256−42, 256, 40537)-Net in Base 2 — Upper bound on s
There is no (214, 256, 40538)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 115806 788560 728038 312218 136296 062080 387005 394422 211993 189734 606435 514852 357686 > 2256 [i]