Best Known (107−18, 107, s)-Nets in Base 8
(107−18, 107, 233017)-Net over F8 — Constructive and digital
Digital (89, 107, 233017)-net over F8, using
- 81 times duplication [i] based on digital (88, 106, 233017)-net over F8, using
- net defined by OOA [i] based on linear OOA(8106, 233017, F8, 18, 18) (dual of [(233017, 18), 4194200, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8106, 2097153, F8, 18) (dual of [2097153, 2097047, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8106, 2097159, F8, 18) (dual of [2097159, 2097053, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(8106, 2097152, F8, 18) (dual of [2097152, 2097046, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(899, 2097152, F8, 17) (dual of [2097152, 2097053, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(8106, 2097159, F8, 18) (dual of [2097159, 2097053, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(8106, 2097153, F8, 18) (dual of [2097153, 2097047, 19]-code), using
- net defined by OOA [i] based on linear OOA(8106, 233017, F8, 18, 18) (dual of [(233017, 18), 4194200, 19]-NRT-code), using
(107−18, 107, 1048580)-Net over F8 — Digital
Digital (89, 107, 1048580)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8107, 1048580, F8, 2, 18) (dual of [(1048580, 2), 2097053, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8107, 2097160, F8, 18) (dual of [2097160, 2097053, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(8106, 2097159, F8, 18) (dual of [2097159, 2097053, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(8106, 2097152, F8, 18) (dual of [2097152, 2097046, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(899, 2097152, F8, 17) (dual of [2097152, 2097053, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(80, 7, F8, 0) (dual of [7, 7, 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(17) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(8106, 2097159, F8, 18) (dual of [2097159, 2097053, 19]-code), using
- OOA 2-folding [i] based on linear OA(8107, 2097160, F8, 18) (dual of [2097160, 2097053, 19]-code), using
(107−18, 107, large)-Net in Base 8 — Upper bound on s
There is no (89, 107, large)-net in base 8, because
- 16 times m-reduction [i] would yield (89, 91, large)-net in base 8, but