Best Known (75, 75+25, s)-Nets in Base 8
(75, 75+25, 683)-Net over F8 — Constructive and digital
Digital (75, 100, 683)-net over F8, using
- 81 times duplication [i] based on digital (74, 99, 683)-net over F8, using
- net defined by OOA [i] based on linear OOA(899, 683, F8, 25, 25) (dual of [(683, 25), 16976, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(899, 8197, F8, 25) (dual of [8197, 8098, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(898, 8196, F8, 25) (dual of [8196, 8098, 26]-code), using
- trace code [i] based on linear OA(6449, 4098, F64, 25) (dual of [4098, 4049, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(6449, 4096, F64, 25) (dual of [4096, 4047, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(6449, 4098, F64, 25) (dual of [4098, 4049, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(898, 8196, F8, 25) (dual of [8196, 8098, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(899, 8197, F8, 25) (dual of [8197, 8098, 26]-code), using
- net defined by OOA [i] based on linear OOA(899, 683, F8, 25, 25) (dual of [(683, 25), 16976, 26]-NRT-code), using
(75, 75+25, 1028)-Net in Base 8 — Constructive
(75, 100, 1028)-net in base 8, using
- base change [i] based on digital (50, 75, 1028)-net over F16, using
- 161 times duplication [i] based on digital (49, 74, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (12, 24, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- digital (25, 50, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 25, 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, 25, 257)-net over F256, using
- digital (12, 24, 514)-net over F16, using
- (u, u+v)-construction [i] based on
- 161 times duplication [i] based on digital (49, 74, 1028)-net over F16, using
(75, 75+25, 8204)-Net over F8 — Digital
Digital (75, 100, 8204)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8100, 8204, F8, 25) (dual of [8204, 8104, 26]-code), using
- trace code [i] based on linear OA(6450, 4102, F64, 25) (dual of [4102, 4052, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(6449, 4097, F64, 25) (dual of [4097, 4048, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(6445, 4097, F64, 23) (dual of [4097, 4052, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,12]) ⊂ C([0,11]) [i] based on
- trace code [i] based on linear OA(6450, 4102, F64, 25) (dual of [4102, 4052, 26]-code), using
(75, 75+25, large)-Net in Base 8 — Upper bound on s
There is no (75, 100, large)-net in base 8, because
- 23 times m-reduction [i] would yield (75, 77, large)-net in base 8, but