Best Known (29, 40, s)-Nets in Base 8
(29, 40, 821)-Net over F8 — Constructive and digital
Digital (29, 40, 821)-net over F8, using
- 81 times duplication [i] based on digital (28, 39, 821)-net over F8, using
- net defined by OOA [i] based on linear OOA(839, 821, F8, 11, 11) (dual of [(821, 11), 8992, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(839, 4106, F8, 11) (dual of [4106, 4067, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(838, 4105, F8, 11) (dual of [4105, 4067, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(837, 4096, F8, 11) (dual of [4096, 4059, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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 Ce(10) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(838, 4105, F8, 11) (dual of [4105, 4067, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(839, 4106, F8, 11) (dual of [4106, 4067, 12]-code), using
- net defined by OOA [i] based on linear OOA(839, 821, F8, 11, 11) (dual of [(821, 11), 8992, 12]-NRT-code), using
(29, 40, 4145)-Net over F8 — Digital
Digital (29, 40, 4145)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(840, 4145, F8, 11) (dual of [4145, 4105, 12]-code), using
- 42 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 34 times 0) [i] based on linear OA(837, 4100, F8, 11) (dual of [4100, 4063, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(837, 4096, F8, 11) (dual of [4096, 4059, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(833, 4096, F8, 10) (dual of [4096, 4063, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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(10) ⊂ Ce(9) [i] based on
- 42 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 34 times 0) [i] based on linear OA(837, 4100, F8, 11) (dual of [4100, 4063, 12]-code), using
(29, 40, 4119455)-Net in Base 8 — Upper bound on s
There is no (29, 40, 4119456)-net in base 8, because
- 1 times m-reduction [i] would yield (29, 39, 4119456)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 166153 589248 272746 802781 360039 467305 > 839 [i]