Best Known (42, 42+24, s)-Nets in Base 8
(42, 42+24, 354)-Net over F8 — Constructive and digital
Digital (42, 66, 354)-net over F8, using
- 4 times m-reduction [i] based on digital (42, 70, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 35, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 35, 177)-net over F64, using
(42, 42+24, 514)-Net in Base 8 — Constructive
(42, 66, 514)-net in base 8, using
- 82 times duplication [i] based on (40, 64, 514)-net in base 8, using
- base change [i] based on digital (24, 48, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 24, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 24, 257)-net over F256, using
- base change [i] based on digital (24, 48, 514)-net over F16, using
(42, 42+24, 545)-Net over F8 — Digital
Digital (42, 66, 545)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(866, 545, F8, 24) (dual of [545, 479, 25]-code), using
- 26 step Varšamov–Edel lengthening with (ri) = (1, 4 times 0, 1, 20 times 0) [i] based on linear OA(864, 517, F8, 24) (dual of [517, 453, 25]-code), using
- construction XX applied to C1 = C([510,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([510,22]) [i] based on
- linear OA(861, 511, F8, 23) (dual of [511, 450, 24]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(861, 511, F8, 23) (dual of [511, 450, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(864, 511, F8, 24) (dual of [511, 447, 25]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(858, 511, F8, 22) (dual of [511, 453, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([510,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([510,22]) [i] based on
- 26 step Varšamov–Edel lengthening with (ri) = (1, 4 times 0, 1, 20 times 0) [i] based on linear OA(864, 517, F8, 24) (dual of [517, 453, 25]-code), using
(42, 42+24, 70018)-Net in Base 8 — Upper bound on s
There is no (42, 66, 70019)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 401760 540233 693384 926938 142831 820000 412963 224826 931230 412064 > 866 [i]