Best Known (209−43, 209, s)-Nets in Base 2
(209−43, 209, 260)-Net over F2 — Constructive and digital
Digital (166, 209, 260)-net over F2, using
- t-expansion [i] based on digital (165, 209, 260)-net over F2, using
- 3 times m-reduction [i] based on digital (165, 212, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 53, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 53, 65)-net over F16, using
- 3 times m-reduction [i] based on digital (165, 212, 260)-net over F2, using
(209−43, 209, 514)-Net over F2 — Digital
Digital (166, 209, 514)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2209, 514, F2, 2, 43) (dual of [(514, 2), 819, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 523, F2, 2, 43) (dual of [(523, 2), 837, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2209, 1046, F2, 43) (dual of [1046, 837, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(2209, 1047, F2, 43) (dual of [1047, 838, 44]-code), using
- adding a parity check bit [i] based on linear OA(2208, 1046, F2, 42) (dual of [1046, 838, 43]-code), using
- construction XX applied to C1 = C([1021,38]), C2 = C([1,40]), C3 = C1 + C2 = C([1,38]), and C∩ = C1 ∩ C2 = C([1021,40]) [i] based on
- linear OA(2196, 1023, F2, 41) (dual of [1023, 827, 42]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,38}, and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(2195, 1023, F2, 40) (dual of [1023, 828, 41]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(2206, 1023, F2, 43) (dual of [1023, 817, 44]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,40}, and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(2185, 1023, F2, 38) (dual of [1023, 838, 39]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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 (see above)
- construction XX applied to C1 = C([1021,38]), C2 = C([1,40]), C3 = C1 + C2 = C([1,38]), and C∩ = C1 ∩ C2 = C([1021,40]) [i] based on
- adding a parity check bit [i] based on linear OA(2208, 1046, F2, 42) (dual of [1046, 838, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(2209, 1047, F2, 43) (dual of [1047, 838, 44]-code), using
- OOA 2-folding [i] based on linear OA(2209, 1046, F2, 43) (dual of [1046, 837, 44]-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 523, F2, 2, 43) (dual of [(523, 2), 837, 44]-NRT-code), using
(209−43, 209, 8289)-Net in Base 2 — Upper bound on s
There is no (166, 209, 8290)-net in base 2, because
- 1 times m-reduction [i] would yield (166, 208, 8290)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 412 367508 274584 347833 897263 907439 533637 936606 942920 103984 234984 > 2208 [i]