Best Known (121, 121+29, s)-Nets in Base 2
(121, 121+29, 260)-Net over F2 — Constructive and digital
Digital (121, 150, 260)-net over F2, using
- t-expansion [i] based on digital (120, 150, 260)-net over F2, using
- 2 times m-reduction [i] based on digital (120, 152, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 38, 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, 38, 65)-net over F16, using
- 2 times m-reduction [i] based on digital (120, 152, 260)-net over F2, using
(121, 121+29, 509)-Net over F2 — Digital
Digital (121, 150, 509)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2150, 509, F2, 2, 29) (dual of [(509, 2), 868, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2150, 531, F2, 2, 29) (dual of [(531, 2), 912, 30]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2149, 531, F2, 2, 29) (dual of [(531, 2), 913, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2149, 1062, F2, 29) (dual of [1062, 913, 30]-code), using
- adding a parity check bit [i] based on linear OA(2148, 1061, F2, 28) (dual of [1061, 913, 29]-code), using
- construction XX applied to C1 = C([1019,22]), C2 = C([1,24]), C3 = C1 + C2 = C([1,22]), and C∩ = C1 ∩ C2 = C([1019,24]) [i] based on
- linear OA(2131, 1023, F2, 27) (dual of [1023, 892, 28]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,22}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2120, 1023, F2, 24) (dual of [1023, 903, 25]-code), using the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2141, 1023, F2, 29) (dual of [1023, 882, 30]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,24}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(2110, 1023, F2, 22) (dual of [1023, 913, 23]-code), using 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(26, 27, F2, 3) (dual of [27, 21, 4]-code or 27-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([1019,22]), C2 = C([1,24]), C3 = C1 + C2 = C([1,22]), and C∩ = C1 ∩ C2 = C([1019,24]) [i] based on
- adding a parity check bit [i] based on linear OA(2148, 1061, F2, 28) (dual of [1061, 913, 29]-code), using
- OOA 2-folding [i] based on linear OA(2149, 1062, F2, 29) (dual of [1062, 913, 30]-code), using
- 21 times duplication [i] based on linear OOA(2149, 531, F2, 2, 29) (dual of [(531, 2), 913, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2150, 531, F2, 2, 29) (dual of [(531, 2), 912, 30]-NRT-code), using
(121, 121+29, 9646)-Net in Base 2 — Upper bound on s
There is no (121, 150, 9647)-net in base 2, because
- 1 times m-reduction [i] would yield (121, 149, 9647)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 714 459843 828314 538429 102591 274796 681209 480993 > 2149 [i]