Best Known (165−33, 165, s)-Nets in Base 2
(165−33, 165, 260)-Net over F2 — Constructive and digital
Digital (132, 165, 260)-net over F2, using
- 3 times m-reduction [i] based on digital (132, 168, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 42, 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, 42, 65)-net over F16, using
(165−33, 165, 479)-Net over F2 — Digital
Digital (132, 165, 479)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2165, 479, F2, 2, 33) (dual of [(479, 2), 793, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2165, 524, F2, 2, 33) (dual of [(524, 2), 883, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2165, 1048, F2, 33) (dual of [1048, 883, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2164, 1047, F2, 33) (dual of [1047, 883, 34]-code), using
- adding a parity check bit [i] based on linear OA(2163, 1046, F2, 32) (dual of [1046, 883, 33]-code), using
- construction XX applied to C1 = C([1021,28]), C2 = C([1,30]), C3 = C1 + C2 = C([1,28]), and C∩ = C1 ∩ C2 = C([1021,30]) [i] based on
- linear OA(2151, 1023, F2, 31) (dual of [1023, 872, 32]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,28}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2150, 1023, F2, 30) (dual of [1023, 873, 31]-code), using 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(2161, 1023, F2, 33) (dual of [1023, 862, 34]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,30}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2140, 1023, F2, 28) (dual of [1023, 883, 29]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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, 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 (see above)
- construction XX applied to C1 = C([1021,28]), C2 = C([1,30]), C3 = C1 + C2 = C([1,28]), and C∩ = C1 ∩ C2 = C([1021,30]) [i] based on
- adding a parity check bit [i] based on linear OA(2163, 1046, F2, 32) (dual of [1046, 883, 33]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2164, 1047, F2, 33) (dual of [1047, 883, 34]-code), using
- OOA 2-folding [i] based on linear OA(2165, 1048, F2, 33) (dual of [1048, 883, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2165, 524, F2, 2, 33) (dual of [(524, 2), 883, 34]-NRT-code), using
(165−33, 165, 8257)-Net in Base 2 — Upper bound on s
There is no (132, 165, 8258)-net in base 2, because
- 1 times m-reduction [i] would yield (132, 164, 8258)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 23 393712 775828 245305 768066 886674 678008 115474 747100 > 2164 [i]