Best Known (161−32, 161, s)-Nets in Base 2
(161−32, 161, 260)-Net over F2 — Constructive and digital
Digital (129, 161, 260)-net over F2, using
- 3 times m-reduction [i] based on digital (129, 164, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 41, 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, 41, 65)-net over F16, using
(161−32, 161, 481)-Net over F2 — Digital
Digital (129, 161, 481)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2161, 481, F2, 2, 32) (dual of [(481, 2), 801, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2161, 517, F2, 2, 32) (dual of [(517, 2), 873, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2161, 1034, F2, 32) (dual of [1034, 873, 33]-code), using
- 1 times truncation [i] based on linear OA(2162, 1035, F2, 33) (dual of [1035, 873, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(30) [i] based on
- linear OA(2161, 1024, F2, 33) (dual of [1024, 863, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2151, 1024, F2, 31) (dual of [1024, 873, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [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(32) ⊂ Ce(30) [i] based on
- 1 times truncation [i] based on linear OA(2162, 1035, F2, 33) (dual of [1035, 873, 34]-code), using
- OOA 2-folding [i] based on linear OA(2161, 1034, F2, 32) (dual of [1034, 873, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(2161, 517, F2, 2, 32) (dual of [(517, 2), 873, 33]-NRT-code), using
(161−32, 161, 7248)-Net in Base 2 — Upper bound on s
There is no (129, 161, 7249)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2 926086 884538 038654 450283 270148 722079 083457 789485 > 2161 [i]