Best Known (89, 89+22, s)-Nets in Base 2
(89, 89+22, 196)-Net over F2 — Constructive and digital
Digital (89, 111, 196)-net over F2, using
- 1 times m-reduction [i] based on digital (89, 112, 196)-net over F2, using
- trace code for nets [i] based on digital (5, 28, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- trace code for nets [i] based on digital (5, 28, 49)-net over F16, using
(89, 89+22, 396)-Net over F2 — Digital
Digital (89, 111, 396)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2111, 396, F2, 2, 22) (dual of [(396, 2), 681, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2111, 517, F2, 2, 22) (dual of [(517, 2), 923, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2111, 1034, F2, 22) (dual of [1034, 923, 23]-code), using
- 1 times truncation [i] based on linear OA(2112, 1035, F2, 23) (dual of [1035, 923, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2111, 1024, F2, 23) (dual of [1024, 913, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2101, 1024, F2, 21) (dual of [1024, 923, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- 1 times truncation [i] based on linear OA(2112, 1035, F2, 23) (dual of [1035, 923, 24]-code), using
- OOA 2-folding [i] based on linear OA(2111, 1034, F2, 22) (dual of [1034, 923, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(2111, 517, F2, 2, 22) (dual of [(517, 2), 923, 23]-NRT-code), using
(89, 89+22, 5338)-Net in Base 2 — Upper bound on s
There is no (89, 111, 5339)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2601 233755 918097 176116 263200 943450 > 2111 [i]