Best Known (23, 23+12, s)-Nets in Base 8
(23, 23+12, 256)-Net over F8 — Constructive and digital
Digital (23, 35, 256)-net over F8, using
- 1 times m-reduction [i] based on digital (23, 36, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 18, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 18, 128)-net over F64, using
(23, 23+12, 514)-Net in Base 8 — Constructive
(23, 35, 514)-net in base 8, using
- 1 times m-reduction [i] based on (23, 36, 514)-net in base 8, using
- base change [i] based on digital (14, 27, 514)-net over F16, using
- 1 times m-reduction [i] based on digital (14, 28, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 14, 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, 14, 257)-net over F256, using
- 1 times m-reduction [i] based on digital (14, 28, 514)-net over F16, using
- base change [i] based on digital (14, 27, 514)-net over F16, using
(23, 23+12, 579)-Net over F8 — Digital
Digital (23, 35, 579)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(835, 579, F8, 12) (dual of [579, 544, 13]-code), using
- 58 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 14 times 0, 1, 37 times 0) [i] based on linear OA(831, 517, F8, 12) (dual of [517, 486, 13]-code), using
- construction XX applied to C1 = C([510,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([510,10]) [i] based on
- linear OA(828, 511, F8, 11) (dual of [511, 483, 12]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,9}, and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(828, 511, F8, 11) (dual of [511, 483, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(831, 511, F8, 12) (dual of [511, 480, 13]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(825, 511, F8, 10) (dual of [511, 486, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [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([510,9]), C2 = C([0,10]), C3 = C1 + C2 = C([0,9]), and C∩ = C1 ∩ C2 = C([510,10]) [i] based on
- 58 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 14 times 0, 1, 37 times 0) [i] based on linear OA(831, 517, F8, 12) (dual of [517, 486, 13]-code), using
(23, 23+12, 79273)-Net in Base 8 — Upper bound on s
There is no (23, 35, 79274)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 40 564929 384392 450740 588455 037924 > 835 [i]