Best Known (57, 57+19, s)-Nets in Base 8
(57, 57+19, 911)-Net over F8 — Constructive and digital
Digital (57, 76, 911)-net over F8, using
- net defined by OOA [i] based on linear OOA(876, 911, F8, 19, 19) (dual of [(911, 19), 17233, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(876, 8200, F8, 19) (dual of [8200, 8124, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(876, 8204, F8, 19) (dual of [8204, 8128, 20]-code), using
- trace code [i] based on linear OA(6438, 4102, F64, 19) (dual of [4102, 4064, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- trace code [i] based on linear OA(6438, 4102, F64, 19) (dual of [4102, 4064, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(876, 8204, F8, 19) (dual of [8204, 8128, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(876, 8200, F8, 19) (dual of [8200, 8124, 20]-code), using
(57, 57+19, 1028)-Net in Base 8 — Constructive
(57, 76, 1028)-net in base 8, using
- base change [i] based on digital (38, 57, 1028)-net over F16, using
- 161 times duplication [i] based on digital (37, 56, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (9, 18, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 9, 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, 9, 257)-net over F256, using
- digital (19, 38, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 19, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 19, 257)-net over F256, using
- digital (9, 18, 514)-net over F16, using
- (u, u+v)-construction [i] based on
- 161 times duplication [i] based on digital (37, 56, 1028)-net over F16, using
(57, 57+19, 8204)-Net over F8 — Digital
Digital (57, 76, 8204)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(876, 8204, F8, 19) (dual of [8204, 8128, 20]-code), using
- trace code [i] based on linear OA(6438, 4102, F64, 19) (dual of [4102, 4064, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(6437, 4097, F64, 19) (dual of [4097, 4060, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(6433, 4097, F64, 17) (dual of [4097, 4064, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- trace code [i] based on linear OA(6438, 4102, F64, 19) (dual of [4102, 4064, 20]-code), using
(57, 57+19, large)-Net in Base 8 — Upper bound on s
There is no (57, 76, large)-net in base 8, because
- 17 times m-reduction [i] would yield (57, 59, large)-net in base 8, but