Best Known (27, 27+15, s)-Nets in Base 8
(27, 27+15, 256)-Net over F8 — Constructive and digital
Digital (27, 42, 256)-net over F8, using
- 2 times m-reduction [i] based on digital (27, 44, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 22, 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, 22, 128)-net over F64, using
(27, 27+15, 514)-Net in Base 8 — Constructive
(27, 42, 514)-net in base 8, using
- 82 times duplication [i] based on (25, 40, 514)-net in base 8, using
- base change [i] based on digital (15, 30, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 15, 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, 15, 257)-net over F256, using
- base change [i] based on digital (15, 30, 514)-net over F16, using
(27, 27+15, 525)-Net over F8 — Digital
Digital (27, 42, 525)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(842, 525, F8, 15) (dual of [525, 483, 16]-code), using
- construction XX applied to C1 = C([509,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([509,12]) [i] based on
- linear OA(834, 511, F8, 13) (dual of [511, 477, 14]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(834, 511, F8, 13) (dual of [511, 477, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(840, 511, F8, 15) (dual of [511, 471, 16]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−2,−1,…,12}, and designed minimum distance d ≥ |I|+1 = 16 [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(81, 7, F8, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 8, F8, 1) (dual of [8, 7, 2]-code), using
- Reed–Solomon code RS(7,8) [i]
- discarding factors / shortening the dual code based on linear OA(81, 8, F8, 1) (dual of [8, 7, 2]-code), using
- linear OA(81, 7, F8, 1) (dual of [7, 6, 2]-code) (see above)
- construction XX applied to C1 = C([509,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([509,12]) [i] based on
(27, 27+15, 94043)-Net in Base 8 — Upper bound on s
There is no (27, 42, 94044)-net in base 8, because
- 1 times m-reduction [i] would yield (27, 41, 94044)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 10 634147 810313 242653 907590 281065 406418 > 841 [i]