Best Known (39, 39+21, s)-Nets in Base 8
(39, 39+21, 354)-Net over F8 — Constructive and digital
Digital (39, 60, 354)-net over F8, using
- 4 times m-reduction [i] based on digital (39, 64, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 32, 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, 32, 177)-net over F64, using
(39, 39+21, 516)-Net in Base 8 — Constructive
(39, 60, 516)-net in base 8, using
- base change [i] based on digital (24, 45, 516)-net over F16, using
- 1 times m-reduction [i] based on digital (24, 46, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 23, 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, 23, 258)-net over F256, using
- 1 times m-reduction [i] based on digital (24, 46, 516)-net over F16, using
(39, 39+21, 621)-Net over F8 — Digital
Digital (39, 60, 621)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(860, 621, F8, 21) (dual of [621, 561, 22]-code), using
- 99 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 13 times 0, 1, 30 times 0, 1, 49 times 0) [i] based on linear OA(855, 517, F8, 21) (dual of [517, 462, 22]-code), using
- construction XX applied to C1 = C([510,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([510,19]) [i] based on
- 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(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(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(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(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([510,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([510,19]) [i] based on
- 99 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 13 times 0, 1, 30 times 0, 1, 49 times 0) [i] based on linear OA(855, 517, F8, 21) (dual of [517, 462, 22]-code), using
(39, 39+21, 137749)-Net in Base 8 — Upper bound on s
There is no (39, 60, 137750)-net in base 8, because
- 1 times m-reduction [i] would yield (39, 59, 137750)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 191566 846565 356696 787983 540391 726734 538904 607338 365676 > 859 [i]