Best Known (129−23, 129, s)-Nets in Base 8
(129−23, 129, 23835)-Net over F8 — Constructive and digital
Digital (106, 129, 23835)-net over F8, using
- 81 times duplication [i] based on digital (105, 128, 23835)-net over F8, using
- net defined by OOA [i] based on linear OOA(8128, 23835, F8, 23, 23) (dual of [(23835, 23), 548077, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8128, 262186, F8, 23) (dual of [262186, 262058, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8128, 262188, F8, 23) (dual of [262188, 262060, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(8121, 262145, F8, 23) (dual of [262145, 262024, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(885, 262145, F8, 17) (dual of [262145, 262060, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(87, 43, F8, 5) (dual of [43, 36, 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(8128, 262188, F8, 23) (dual of [262188, 262060, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8128, 262186, F8, 23) (dual of [262186, 262058, 24]-code), using
- net defined by OOA [i] based on linear OOA(8128, 23835, F8, 23, 23) (dual of [(23835, 23), 548077, 24]-NRT-code), using
(129−23, 129, 262190)-Net over F8 — Digital
Digital (106, 129, 262190)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8129, 262190, F8, 23) (dual of [262190, 262061, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8128, 262188, F8, 23) (dual of [262188, 262060, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(8121, 262145, F8, 23) (dual of [262145, 262024, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(885, 262145, F8, 17) (dual of [262145, 262060, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 812−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(87, 43, F8, 5) (dual of [43, 36, 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(8128, 262189, F8, 22) (dual of [262189, 262061, 23]-code), using Gilbert–Varšamov bound and bm = 8128 > Vbs−1(k−1) = 6749 025443 628099 510372 654988 035308 115830 934117 715866 198242 501109 679326 114628 469194 238548 162626 151058 843720 371708 [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(8128, 262188, F8, 23) (dual of [262188, 262060, 24]-code), using
- construction X with Varšamov bound [i] based on
(129−23, 129, large)-Net in Base 8 — Upper bound on s
There is no (106, 129, large)-net in base 8, because
- 21 times m-reduction [i] would yield (106, 108, large)-net in base 8, but