Best Known (108, 108+21, s)-Nets in Base 8
(108, 108+21, 209716)-Net over F8 — Constructive and digital
Digital (108, 129, 209716)-net over F8, using
- 81 times duplication [i] based on digital (107, 128, 209716)-net over F8, using
- net defined by OOA [i] based on linear OOA(8128, 209716, F8, 21, 21) (dual of [(209716, 21), 4403908, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8128, 2097161, F8, 21) (dual of [2097161, 2097033, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8128, 2097168, F8, 21) (dual of [2097168, 2097040, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(8127, 2097153, F8, 21) (dual of [2097153, 2097026, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(8113, 2097153, F8, 19) (dual of [2097153, 2097040, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(81, 15, F8, 1) (dual of [15, 14, 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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8128, 2097168, F8, 21) (dual of [2097168, 2097040, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8128, 2097161, F8, 21) (dual of [2097161, 2097033, 22]-code), using
- net defined by OOA [i] based on linear OOA(8128, 209716, F8, 21, 21) (dual of [(209716, 21), 4403908, 22]-NRT-code), using
(108, 108+21, 1374322)-Net over F8 — Digital
Digital (108, 129, 1374322)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8129, 1374322, F8, 21) (dual of [1374322, 1374193, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8129, 2097169, F8, 21) (dual of [2097169, 2097040, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(8128, 2097168, F8, 21) (dual of [2097168, 2097040, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(8127, 2097153, F8, 21) (dual of [2097153, 2097026, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(8113, 2097153, F8, 19) (dual of [2097153, 2097040, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(81, 15, F8, 1) (dual of [15, 14, 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 C([0,10]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(8128, 2097168, F8, 21) (dual of [2097168, 2097040, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8129, 2097169, F8, 21) (dual of [2097169, 2097040, 22]-code), using
(108, 108+21, large)-Net in Base 8 — Upper bound on s
There is no (108, 129, large)-net in base 8, because
- 19 times m-reduction [i] would yield (108, 110, large)-net in base 8, but