Best Known (39−11, 39, s)-Nets in Base 8
(39−11, 39, 821)-Net over F8 — Constructive and digital
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
(39−11, 39, 3847)-Net over F8 — Digital
Digital (28, 39, 3847)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(839, 3847, F8, 11) (dual of [3847, 3808, 12]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(839, 4106, F8, 11) (dual of [4106, 4067, 12]-code), using
(39−11, 39, 2717826)-Net in Base 8 — Upper bound on s
There is no (28, 39, 2717827)-net in base 8, because
- 1 times m-reduction [i] would yield (28, 38, 2717827)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 20769 219040 022146 402206 252136 268608 > 838 [i]