Best Known (50, 50+19, s)-Nets in Base 8
(50, 50+19, 456)-Net over F8 — Constructive and digital
Digital (50, 69, 456)-net over F8, using
- 83 times duplication [i] based on digital (47, 66, 456)-net over F8, using
- net defined by OOA [i] based on linear OOA(866, 456, F8, 19, 19) (dual of [(456, 19), 8598, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(866, 4105, F8, 19) (dual of [4105, 4039, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(866, 4106, F8, 19) (dual of [4106, 4040, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(865, 4097, F8, 19) (dual of [4097, 4032, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 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
- discarding factors / shortening the dual code based on linear OA(866, 4106, F8, 19) (dual of [4106, 4040, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(866, 4105, F8, 19) (dual of [4105, 4039, 20]-code), using
- net defined by OOA [i] based on linear OOA(866, 456, F8, 19, 19) (dual of [(456, 19), 8598, 20]-NRT-code), using
(50, 50+19, 579)-Net in Base 8 — Constructive
(50, 69, 579)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (8, 17, 65)-net over F8, using
- base reduction for projective spaces (embedding PG(8,64) in PG(16,8)) for nets [i] based on digital (0, 9, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- base reduction for projective spaces (embedding PG(8,64) in PG(16,8)) for nets [i] based on digital (0, 9, 65)-net over F64, using
- (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
- base change [i] based on digital (20, 39, 514)-net over F16, using
- digital (8, 17, 65)-net over F8, using
(50, 50+19, 4131)-Net over F8 — Digital
Digital (50, 69, 4131)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(869, 4131, F8, 19) (dual of [4131, 4062, 20]-code), using
- 22 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 16 times 0) [i] based on linear OA(866, 4106, F8, 19) (dual of [4106, 4040, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(865, 4097, F8, 19) (dual of [4097, 4032, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(857, 4097, F8, 17) (dual of [4097, 4040, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 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
- 22 step Varšamov–Edel lengthening with (ri) = (2, 4 times 0, 1, 16 times 0) [i] based on linear OA(866, 4106, F8, 19) (dual of [4106, 4040, 20]-code), using
(50, 50+19, 3944567)-Net in Base 8 — Upper bound on s
There is no (50, 69, 3944568)-net in base 8, because
- 1 times m-reduction [i] would yield (50, 68, 3944568)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 25 711036 790263 511600 187036 171310 318682 922094 807697 662999 241606 > 868 [i]