Best Known (34, 34+10, s)-Nets in Base 2
(34, 34+10, 96)-Net over F2 — Constructive and digital
Digital (34, 44, 96)-net over F2, using
- trace code for nets [i] based on digital (1, 11, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
(34, 34+10, 146)-Net over F2 — Digital
Digital (34, 44, 146)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(244, 146, F2, 10) (dual of [146, 102, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 275, F2, 10) (dual of [275, 231, 11]-code), using
- strength reduction [i] based on linear OA(244, 275, F2, 11) (dual of [275, 231, 12]-code), using
- adding a parity check bit [i] based on linear OA(243, 274, F2, 10) (dual of [274, 231, 11]-code), using
- construction XX applied to C1 = C([253,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), and C∩ = C1 ∩ C2 = C([253,8]) [i] based on
- linear OA(233, 255, F2, 9) (dual of [255, 222, 10]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,6}, and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(232, 255, F2, 8) (dual of [255, 223, 9]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(241, 255, F2, 11) (dual of [255, 214, 12]-code), using the primitive BCH-code C(I) with length 255 = 28−1, defining interval I = {−2,−1,…,8}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(224, 255, F2, 6) (dual of [255, 231, 7]-code), using the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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, 9, F2, 1) (dual of [9, 8, 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([253,6]), C2 = C([1,8]), C3 = C1 + C2 = C([1,6]), and C∩ = C1 ∩ C2 = C([253,8]) [i] based on
- adding a parity check bit [i] based on linear OA(243, 274, F2, 10) (dual of [274, 231, 11]-code), using
- strength reduction [i] based on linear OA(244, 275, F2, 11) (dual of [275, 231, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(244, 275, F2, 10) (dual of [275, 231, 11]-code), using
(34, 34+10, 1154)-Net in Base 2 — Upper bound on s
There is no (34, 44, 1155)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 17 652885 481232 > 244 [i]