Best Known (131−26, 131, s)-Nets in Base 2
(131−26, 131, 260)-Net over F2 — Constructive and digital
Digital (105, 131, 260)-net over F2, using
- 1 times m-reduction [i] based on digital (105, 132, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 33, 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, 33, 65)-net over F16, using
(131−26, 131, 428)-Net over F2 — Digital
Digital (105, 131, 428)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2131, 428, F2, 2, 26) (dual of [(428, 2), 725, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2131, 517, F2, 2, 26) (dual of [(517, 2), 903, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2131, 1034, F2, 26) (dual of [1034, 903, 27]-code), using
- 1 times truncation [i] based on linear OA(2132, 1035, F2, 27) (dual of [1035, 903, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2131, 1024, F2, 27) (dual of [1024, 893, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2121, 1024, F2, 25) (dual of [1024, 903, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(26) ⊂ Ce(24) [i] based on
- 1 times truncation [i] based on linear OA(2132, 1035, F2, 27) (dual of [1035, 903, 28]-code), using
- OOA 2-folding [i] based on linear OA(2131, 1034, F2, 26) (dual of [1034, 903, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(2131, 517, F2, 2, 26) (dual of [(517, 2), 903, 27]-NRT-code), using
(131−26, 131, 6102)-Net in Base 2 — Upper bound on s
There is no (105, 131, 6103)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2724 635429 322406 470206 177241 867811 387592 > 2131 [i]