Best Known (42, 64, s)-Nets in Base 8
(42, 64, 354)-Net over F8 — Constructive and digital
Digital (42, 64, 354)-net over F8, using
- 6 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, 64, 518)-Net in Base 8 — Constructive
(42, 64, 518)-net in base 8, using
- base change [i] based on digital (26, 48, 518)-net over F16, using
- trace code for nets [i] based on digital (2, 24, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- trace code for nets [i] based on digital (2, 24, 259)-net over F256, using
(42, 64, 712)-Net over F8 — Digital
Digital (42, 64, 712)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(864, 712, F8, 22) (dual of [712, 648, 23]-code), using
- 189 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 20 times 0, 1, 41 times 0, 1, 54 times 0, 1, 64 times 0) [i] based on linear OA(858, 517, F8, 22) (dual of [517, 459, 23]-code), using
- construction XX applied to C1 = C([510,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([510,20]) [i] based on
- linear OA(855, 511, F8, 21) (dual of [511, 456, 22]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(855, 511, F8, 21) (dual of [511, 456, 22]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(858, 511, F8, 22) (dual of [511, 453, 23]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,20}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(852, 511, F8, 20) (dual of [511, 459, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [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([510,19]), C2 = C([0,20]), C3 = C1 + C2 = C([0,19]), and C∩ = C1 ∩ C2 = C([510,20]) [i] based on
- 189 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 20 times 0, 1, 41 times 0, 1, 54 times 0, 1, 64 times 0) [i] based on linear OA(858, 517, F8, 22) (dual of [517, 459, 23]-code), using
(42, 64, 125960)-Net in Base 8 — Upper bound on s
There is no (42, 64, 125961)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 6277 244425 992858 847438 932365 468449 344294 422195 469360 498176 > 864 [i]