Best Known (120, 120+23, s)-Nets in Base 8
(120, 120+23, 190651)-Net over F8 — Constructive and digital
Digital (120, 143, 190651)-net over F8, using
- 81 times duplication [i] based on digital (119, 142, 190651)-net over F8, using
- net defined by OOA [i] based on linear OOA(8142, 190651, F8, 23, 23) (dual of [(190651, 23), 4384831, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8142, 2097162, F8, 23) (dual of [2097162, 2097020, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8142, 2097168, F8, 23) (dual of [2097168, 2097026, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8141, 2097153, F8, 23) (dual of [2097153, 2097012, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- 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(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([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8142, 2097168, F8, 23) (dual of [2097168, 2097026, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8142, 2097162, F8, 23) (dual of [2097162, 2097020, 24]-code), using
- net defined by OOA [i] based on linear OOA(8142, 190651, F8, 23, 23) (dual of [(190651, 23), 4384831, 24]-NRT-code), using
(120, 120+23, 1584892)-Net over F8 — Digital
Digital (120, 143, 1584892)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8143, 1584892, F8, 23) (dual of [1584892, 1584749, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8143, 2097169, F8, 23) (dual of [2097169, 2097026, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(8142, 2097168, F8, 23) (dual of [2097168, 2097026, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8141, 2097153, F8, 23) (dual of [2097153, 2097012, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- 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(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([0,11]) ⊂ C([0,10]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(8142, 2097168, F8, 23) (dual of [2097168, 2097026, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8143, 2097169, F8, 23) (dual of [2097169, 2097026, 24]-code), using
(120, 120+23, large)-Net in Base 8 — Upper bound on s
There is no (120, 143, large)-net in base 8, because
- 21 times m-reduction [i] would yield (120, 122, large)-net in base 8, but