Best Known (123, 123+23, s)-Nets in Base 8
(123, 123+23, 190653)-Net over F8 — Constructive and digital
Digital (123, 146, 190653)-net over F8, using
- 81 times duplication [i] based on digital (122, 145, 190653)-net over F8, using
- net defined by OOA [i] based on linear OOA(8145, 190653, F8, 23, 23) (dual of [(190653, 23), 4384874, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8145, 2097184, F8, 23) (dual of [2097184, 2097039, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, 2097185, F8, 23) (dual of [2097185, 2097040, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [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(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(84, 32, F8, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,8)), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8145, 2097185, F8, 23) (dual of [2097185, 2097040, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8145, 2097184, F8, 23) (dual of [2097184, 2097039, 24]-code), using
- net defined by OOA [i] based on linear OOA(8145, 190653, F8, 23, 23) (dual of [(190653, 23), 4384874, 24]-NRT-code), using
(123, 123+23, 2097187)-Net over F8 — Digital
Digital (123, 146, 2097187)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8146, 2097187, F8, 23) (dual of [2097187, 2097041, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8145, 2097185, F8, 23) (dual of [2097185, 2097040, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [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(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(84, 32, F8, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,8)), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- linear OA(8145, 2097186, F8, 22) (dual of [2097186, 2097041, 23]-code), using Gilbert–Varšamov bound and bm = 8145 > Vbs−1(k−1) = 62093 145466 702123 406103 014560 578405 360638 098480 021164 766781 354407 003667 699228 959687 395769 078965 316525 242992 250299 145786 904866 792184 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 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(8145, 2097185, F8, 23) (dual of [2097185, 2097040, 24]-code), using
- construction X with Varšamov bound [i] based on
(123, 123+23, large)-Net in Base 8 — Upper bound on s
There is no (123, 146, large)-net in base 8, because
- 21 times m-reduction [i] would yield (123, 125, large)-net in base 8, but