Best Known (27, 27+11, s)-Nets in Base 8
(27, 27+11, 820)-Net over F8 — Constructive and digital
Digital (27, 38, 820)-net over F8, using
- net defined by OOA [i] based on linear OOA(838, 820, F8, 11, 11) (dual of [(820, 11), 8982, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(838, 4101, F8, 11) (dual of [4101, 4063, 12]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code 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(838, 4101, F8, 11) (dual of [4101, 4063, 12]-code), using
(27, 27+11, 3053)-Net over F8 — Digital
Digital (27, 38, 3053)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(838, 3053, F8, 11) (dual of [3053, 3015, 12]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(838, 4105, F8, 11) (dual of [4105, 4067, 12]-code), using
(27, 27+11, 1793095)-Net in Base 8 — Upper bound on s
There is no (27, 38, 1793096)-net in base 8, because
- 1 times m-reduction [i] would yield (27, 37, 1793096)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 2596 151006 116726 876286 411374 853051 > 837 [i]