Best Known (257−41, 257, s)-Nets in Base 2
(257−41, 257, 520)-Net over F2 — Constructive and digital
Digital (216, 257, 520)-net over F2, using
- 22 times duplication [i] based on digital (214, 255, 520)-net over F2, using
- t-expansion [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
- t-expansion [i] based on digital (213, 255, 520)-net over F2, using
(257−41, 257, 1519)-Net over F2 — Digital
Digital (216, 257, 1519)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2257, 1519, F2, 2, 41) (dual of [(1519, 2), 2781, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2257, 2074, F2, 2, 41) (dual of [(2074, 2), 3891, 42]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2253, 2072, F2, 2, 41) (dual of [(2072, 2), 3891, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2253, 4144, F2, 41) (dual of [4144, 3891, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- 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(2205, 4096, F2, 35) (dual of [4096, 3891, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- OOA 2-folding [i] based on linear OA(2253, 4144, F2, 41) (dual of [4144, 3891, 42]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2253, 2072, F2, 2, 41) (dual of [(2072, 2), 3891, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2257, 2074, F2, 2, 41) (dual of [(2074, 2), 3891, 42]-NRT-code), using
(257−41, 257, 59193)-Net in Base 2 — Upper bound on s
There is no (216, 257, 59194)-net in base 2, because
- 1 times m-reduction [i] would yield (216, 256, 59194)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 115812 596234 815466 353689 668923 992613 631409 169384 931701 954581 399001 617270 161102 > 2256 [i]