Best Known (128, 128+23, s)-Nets in Base 8
(128, 128+23, 190654)-Net over F8 — Constructive and digital
Digital (128, 151, 190654)-net over F8, using
- 83 times duplication [i] based on digital (125, 148, 190654)-net over F8, using
- net defined by OOA [i] based on linear OOA(8148, 190654, F8, 23, 23) (dual of [(190654, 23), 4384894, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8148, 2097195, F8, 23) (dual of [2097195, 2097047, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8148, 2097202, F8, 23) (dual of [2097202, 2097054, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [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(899, 2097153, F8, 17) (dual of [2097153, 2097054, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(87, 49, F8, 5) (dual of [49, 42, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8148, 2097202, F8, 23) (dual of [2097202, 2097054, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8148, 2097195, F8, 23) (dual of [2097195, 2097047, 24]-code), using
- net defined by OOA [i] based on linear OOA(8148, 190654, F8, 23, 23) (dual of [(190654, 23), 4384894, 24]-NRT-code), using
(128, 128+23, 2097208)-Net over F8 — Digital
Digital (128, 151, 2097208)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8151, 2097208, F8, 23) (dual of [2097208, 2097057, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8148, 2097202, F8, 23) (dual of [2097202, 2097054, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [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(899, 2097153, F8, 17) (dual of [2097153, 2097054, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(87, 49, F8, 5) (dual of [49, 42, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(8148, 2097205, F8, 22) (dual of [2097205, 2097057, 23]-code), using Gilbert–Varšamov bound and bm = 8148 > Vbs−1(k−1) = 62104 960125 976760 094770 275671 035664 084371 943340 295619 931524 476808 619335 971979 757637 901299 960722 275266 073902 770562 920253 728419 415096 [i]
- linear OA(80, 3, F8, 0) (dual of [3, 3, 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
- linear OA(8148, 2097202, F8, 23) (dual of [2097202, 2097054, 24]-code), using
- construction X with Varšamov bound [i] based on
(128, 128+23, large)-Net in Base 8 — Upper bound on s
There is no (128, 151, large)-net in base 8, because
- 21 times m-reduction [i] would yield (128, 130, large)-net in base 8, but