Best Known (52−19, 52, s)-Nets in Base 8
(52−19, 52, 354)-Net over F8 — Constructive and digital
Digital (33, 52, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 26, 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
(52−19, 52, 514)-Net in Base 8 — Constructive
(33, 52, 514)-net in base 8, using
- base change [i] based on digital (20, 39, 514)-net over F16, using
- 1 times m-reduction [i] based on digital (20, 40, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 257)-net over F256, using
- 1 times m-reduction [i] based on digital (20, 40, 514)-net over F16, using
(52−19, 52, 516)-Net over F8 — Digital
Digital (33, 52, 516)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(852, 516, F8, 19) (dual of [516, 464, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(852, 524, F8, 19) (dual of [524, 472, 20]-code), using
- construction XX applied to C1 = C([56,73]), C2 = C([60,74]), C3 = C1 + C2 = C([60,73]), and C∩ = C1 ∩ C2 = C([56,74]) [i] based on
- linear OA(846, 511, F8, 18) (dual of [511, 465, 19]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {56,57,…,73}, and designed minimum distance d ≥ |I|+1 = 19 [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 = {60,61,…,74}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- 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 = {56,57,…,74}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(837, 511, F8, 14) (dual of [511, 474, 15]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {60,61,…,73}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(83, 10, F8, 3) (dual of [10, 7, 4]-code or 10-arc in PG(2,8) or 10-cap in PG(2,8)), using
- 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
- construction XX applied to C1 = C([56,73]), C2 = C([60,74]), C3 = C1 + C2 = C([60,73]), and C∩ = C1 ∩ C2 = C([56,74]) [i] based on
- discarding factors / shortening the dual code based on linear OA(852, 524, F8, 19) (dual of [524, 472, 20]-code), using
(52−19, 52, 77648)-Net in Base 8 — Upper bound on s
There is no (33, 52, 77649)-net in base 8, because
- 1 times m-reduction [i] would yield (33, 51, 77649)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 11418 477091 805346 879808 416253 338575 367786 151168 > 851 [i]