Best Known (207−43, 207, s)-Nets in Base 2
(207−43, 207, 260)-Net over F2 — Constructive and digital
Digital (164, 207, 260)-net over F2, using
- t-expansion [i] based on digital (162, 207, 260)-net over F2, using
- 1 times m-reduction [i] based on digital (162, 208, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 52, 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, 52, 65)-net over F16, using
- 1 times m-reduction [i] based on digital (162, 208, 260)-net over F2, using
(207−43, 207, 495)-Net over F2 — Digital
Digital (164, 207, 495)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2207, 495, F2, 2, 43) (dual of [(495, 2), 783, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2207, 517, F2, 2, 43) (dual of [(517, 2), 827, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2207, 1034, F2, 43) (dual of [1034, 827, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(2207, 1035, F2, 43) (dual of [1035, 828, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(40) [i] based on
- linear OA(2206, 1024, F2, 43) (dual of [1024, 818, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(2196, 1024, F2, 41) (dual of [1024, 828, 42]-code), using an extension Ce(40) of 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(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, using
- construction X applied to Ce(42) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(2207, 1035, F2, 43) (dual of [1035, 828, 44]-code), using
- OOA 2-folding [i] based on linear OA(2207, 1034, F2, 43) (dual of [1034, 827, 44]-code), using
- discarding factors / shortening the dual code based on linear OOA(2207, 517, F2, 2, 43) (dual of [(517, 2), 827, 44]-NRT-code), using
(207−43, 207, 7757)-Net in Base 2 — Upper bound on s
There is no (164, 207, 7758)-net in base 2, because
- 1 times m-reduction [i] would yield (164, 206, 7758)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 102 968912 953700 396384 398161 319925 648703 778848 330347 311919 215732 > 2206 [i]