Best Known (57−20, 57, s)-Nets in Base 8
(57−20, 57, 354)-Net over F8 — Constructive and digital
Digital (37, 57, 354)-net over F8, using
- 3 times m-reduction [i] based on digital (37, 60, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 30, 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, 30, 177)-net over F64, using
(57−20, 57, 516)-Net in Base 8 — Constructive
(37, 57, 516)-net in base 8, using
- 81 times duplication [i] based on (36, 56, 516)-net in base 8, using
- base change [i] based on digital (22, 42, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 21, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 21, 258)-net over F256, using
- base change [i] based on digital (22, 42, 516)-net over F16, using
(57−20, 57, 598)-Net over F8 — Digital
Digital (37, 57, 598)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(857, 598, F8, 20) (dual of [598, 541, 21]-code), using
- 76 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 41 times 0) [i] based on linear OA(852, 517, F8, 20) (dual of [517, 465, 21]-code), using
- construction XX applied to C1 = C([510,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([510,18]) [i] based on
- linear OA(849, 511, F8, 19) (dual of [511, 462, 20]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(849, 511, F8, 19) (dual of [511, 462, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(852, 511, F8, 20) (dual of [511, 459, 21]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(846, 511, F8, 18) (dual of [511, 465, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [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([510,17]), C2 = C([0,18]), C3 = C1 + C2 = C([0,17]), and C∩ = C1 ∩ C2 = C([510,18]) [i] based on
- 76 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 8 times 0, 1, 21 times 0, 1, 41 times 0) [i] based on linear OA(852, 517, F8, 20) (dual of [517, 465, 21]-code), using
(57−20, 57, 90878)-Net in Base 8 — Upper bound on s
There is no (37, 57, 90879)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 2993 202253 618484 186959 965862 520192 333978 711738 763739 > 857 [i]