Best Known (99−25, 99, s)-Nets in Base 8
(99−25, 99, 683)-Net over F8 — Constructive and digital
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
(99−25, 99, 814)-Net in Base 8 — Constructive
(74, 99, 814)-net in base 8, using
- 81 times duplication [i] based on (73, 98, 814)-net in base 8, using
- trace code for nets [i] based on (24, 49, 407)-net in base 64, using
- base change [i] based on (17, 42, 407)-net in base 128, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- (4, 29, 257)-net in base 128, using
- 3 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
- base change [i] based on digital (0, 28, 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, 28, 257)-net over F256, using
- 3 times m-reduction [i] based on (4, 32, 257)-net in base 128, using
- digital (1, 13, 150)-net over F128, using
- (u, u+v)-construction [i] based on
- base change [i] based on (17, 42, 407)-net in base 128, using
- trace code for nets [i] based on (24, 49, 407)-net in base 64, using
(99−25, 99, 8198)-Net over F8 — Digital
Digital (74, 99, 8198)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(899, 8198, F8, 25) (dual of [8198, 8099, 26]-code), using
- construction X with Varšamov bound [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
- linear OA(898, 8197, F8, 24) (dual of [8197, 8099, 25]-code), using Gilbert–Varšamov bound and bm = 898 > Vbs−1(k−1) = 1057 697660 867625 201199 791430 148465 878654 408009 926046 102126 324849 486775 664006 889012 348928 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 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
- linear OA(898, 8196, F8, 25) (dual of [8196, 8098, 26]-code), using
- construction X with Varšamov bound [i] based on
(99−25, 99, large)-Net in Base 8 — Upper bound on s
There is no (74, 99, large)-net in base 8, because
- 23 times m-reduction [i] would yield (74, 76, large)-net in base 8, but