Best Known (69−16, 69, s)-Nets in Base 8
(69−16, 69, 1026)-Net over F8 — Constructive and digital
Digital (53, 69, 1026)-net over F8, using
- 83 times duplication [i] based on digital (50, 66, 1026)-net over F8, using
- net defined by OOA [i] based on linear OOA(866, 1026, F8, 16, 16) (dual of [(1026, 16), 16350, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(866, 8208, F8, 16) (dual of [8208, 8142, 17]-code), using
- trace code [i] based on linear OA(6433, 4104, F64, 16) (dual of [4104, 4071, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- linear OA(6431, 4096, F64, 16) (dual of [4096, 4065, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(6425, 4096, F64, 13) (dual of [4096, 4071, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(15) ⊂ Ce(12) [i] based on
- trace code [i] based on linear OA(6433, 4104, F64, 16) (dual of [4104, 4071, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(866, 8208, F8, 16) (dual of [8208, 8142, 17]-code), using
- net defined by OOA [i] based on linear OOA(866, 1026, F8, 16, 16) (dual of [(1026, 16), 16350, 17]-NRT-code), using
(69−16, 69, 1030)-Net in Base 8 — Constructive
(53, 69, 1030)-net in base 8, using
- 1 times m-reduction [i] based on (53, 70, 1030)-net in base 8, using
- (u, u+v)-construction [i] based on
- (14, 22, 514)-net in base 8, using
- trace code for nets [i] based on (3, 11, 257)-net in base 64, using
- 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
- base change [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
- base change [i] based on digital (0, 9, 257)-net over F256, using
- 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
- trace code for nets [i] based on (3, 11, 257)-net in base 64, using
- (31, 48, 516)-net in base 8, using
- base change [i] based on digital (19, 36, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 18, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 18, 258)-net over F256, using
- base change [i] based on digital (19, 36, 516)-net over F16, using
- (14, 22, 514)-net in base 8, using
- (u, u+v)-construction [i] based on
(69−16, 69, 13096)-Net over F8 — Digital
Digital (53, 69, 13096)-net over F8, using
(69−16, 69, large)-Net in Base 8 — Upper bound on s
There is no (53, 69, large)-net in base 8, because
- 14 times m-reduction [i] would yield (53, 55, large)-net in base 8, but