Best Known (135, 168, s)-Nets in Base 2
(135, 168, 260)-Net over F2 — Constructive and digital
Digital (135, 168, 260)-net over F2, using
- 4 times m-reduction [i] based on digital (135, 172, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 43, 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, 43, 65)-net over F16, using
(135, 168, 516)-Net over F2 — Digital
Digital (135, 168, 516)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2168, 516, F2, 2, 33) (dual of [(516, 2), 864, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2168, 530, F2, 2, 33) (dual of [(530, 2), 892, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2168, 1060, F2, 33) (dual of [1060, 892, 34]-code), using
- construction XX applied to C1 = C([1019,26]), C2 = C([0,28]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([1019,28]) [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 = {−4,−3,…,26}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2141, 1023, F2, 29) (dual of [1023, 882, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [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 = {−4,−3,…,28}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2131, 1023, F2, 27) (dual of [1023, 892, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), 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, using
- construction XX applied to C1 = C([1019,26]), C2 = C([0,28]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([1019,28]) [i] based on
- OOA 2-folding [i] based on linear OA(2168, 1060, F2, 33) (dual of [1060, 892, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2168, 530, F2, 2, 33) (dual of [(530, 2), 892, 34]-NRT-code), using
(135, 168, 9407)-Net in Base 2 — Upper bound on s
There is no (135, 168, 9408)-net in base 2, because
- 1 times m-reduction [i] would yield (135, 167, 9408)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 187 349768 695658 342100 454813 083016 071521 834805 547005 > 2167 [i]