Best Known (81, 81+22, s)-Nets in Base 2
(81, 81+22, 152)-Net over F2 — Constructive and digital
Digital (81, 103, 152)-net over F2, using
- 1 times m-reduction [i] based on digital (81, 104, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 26, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 26, 38)-net over F16, using
(81, 81+22, 266)-Net over F2 — Digital
Digital (81, 103, 266)-net over F2, using
- 21 times duplication [i] based on digital (80, 102, 266)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2102, 266, F2, 2, 22) (dual of [(266, 2), 430, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2102, 532, F2, 22) (dual of [532, 430, 23]-code), using
- construction XX applied to C1 = C([509,18]), C2 = C([1,20]), C3 = C1 + C2 = C([1,18]), and C∩ = C1 ∩ C2 = C([509,20]) [i] based on
- linear OA(291, 511, F2, 21) (dual of [511, 420, 22]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,18}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(290, 511, F2, 20) (dual of [511, 421, 21]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2100, 511, F2, 23) (dual of [511, 411, 24]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,20}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(281, 511, F2, 18) (dual of [511, 430, 19]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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
- 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 (see above)
- construction XX applied to C1 = C([509,18]), C2 = C([1,20]), C3 = C1 + C2 = C([1,18]), and C∩ = C1 ∩ C2 = C([509,20]) [i] based on
- OOA 2-folding [i] based on linear OA(2102, 532, F2, 22) (dual of [532, 430, 23]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2102, 266, F2, 2, 22) (dual of [(266, 2), 430, 23]-NRT-code), using
(81, 81+22, 3218)-Net in Base 2 — Upper bound on s
There is no (81, 103, 3219)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 10 172063 420681 330851 695430 681700 > 2103 [i]