Best Known (29, 36, s)-Nets in Base 8
(29, 36, 11132)-Net over F8 — Constructive and digital
Digital (29, 36, 11132)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 208)-net over F8, using
- net defined by OOA [i] based on linear OOA(85, 208, F8, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(85, 208, F8, 2, 3) (dual of [(208, 2), 411, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(85, 208, F8, 3, 3) (dual of [(208, 3), 619, 4]-NRT-code), using
- digital (24, 31, 10924)-net over F8, using
- net defined by OOA [i] based on linear OOA(831, 10924, F8, 7, 7) (dual of [(10924, 7), 76437, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(831, 32773, F8, 7) (dual of [32773, 32742, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(831, 32768, F8, 7) (dual of [32768, 32737, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(826, 32768, F8, 6) (dual of [32768, 32742, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(80, 5, F8, 0) (dual of [5, 5, 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
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(831, 32773, F8, 7) (dual of [32773, 32742, 8]-code), using
- net defined by OOA [i] based on linear OOA(831, 10924, F8, 7, 7) (dual of [(10924, 7), 76437, 8]-NRT-code), using
- digital (2, 5, 208)-net over F8, using
(29, 36, 43692)-Net in Base 8 — Constructive
(29, 36, 43692)-net in base 8, using
- base change [i] based on digital (20, 27, 43692)-net over F16, using
- net defined by OOA [i] based on linear OOA(1627, 43692, F16, 7, 7) (dual of [(43692, 7), 305817, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1627, 131077, F16, 7) (dual of [131077, 131050, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1626, 131076, F16, 7) (dual of [131076, 131050, 8]-code), using
- trace code [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1626, 131076, F16, 7) (dual of [131076, 131050, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1627, 131077, F16, 7) (dual of [131077, 131050, 8]-code), using
- net defined by OOA [i] based on linear OOA(1627, 43692, F16, 7, 7) (dual of [(43692, 7), 305817, 8]-NRT-code), using
(29, 36, 112118)-Net over F8 — Digital
Digital (29, 36, 112118)-net over F8, using
(29, 36, 131078)-Net in Base 8
(29, 36, 131078)-net in base 8, using
- base change [i] based on digital (20, 27, 131078)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1627, 131078, F16, 7) (dual of [131078, 131051, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1626, 131076, F16, 7) (dual of [131076, 131050, 8]-code), using
- trace code [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(25613, 65536, F256, 7) (dual of [65536, 65523, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(25613, 65538, F256, 7) (dual of [65538, 65525, 8]-code), using
- linear OA(1626, 131077, F16, 6) (dual of [131077, 131051, 7]-code), using Gilbert–Varšamov bound and bm = 1626 > Vbs−1(k−1) = 244826 778925 460333 687782 998016 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(1626, 131076, F16, 7) (dual of [131076, 131050, 8]-code), using
- construction X with Varšamov bound [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1627, 131078, F16, 7) (dual of [131078, 131051, 8]-code), using
(29, 36, large)-Net in Base 8 — Upper bound on s
There is no (29, 36, large)-net in base 8, because
- 5 times m-reduction [i] would yield (29, 31, large)-net in base 8, but