Best Known (114, 114+23, s)-Nets in Base 9
(114, 114+23, 96627)-Net over F9 — Constructive and digital
Digital (114, 137, 96627)-net over F9, using
- 91 times duplication [i] based on digital (113, 136, 96627)-net over F9, using
- net defined by OOA [i] based on linear OOA(9136, 96627, F9, 23, 23) (dual of [(96627, 23), 2222285, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9136, 1062898, F9, 23) (dual of [1062898, 1062762, 24]-code), using
- trace code [i] based on linear OA(8168, 531449, F81, 23) (dual of [531449, 531381, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8167, 531442, F81, 23) (dual of [531442, 531375, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8161, 531442, F81, 21) (dual of [531442, 531381, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 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
- trace code [i] based on linear OA(8168, 531449, F81, 23) (dual of [531449, 531381, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9136, 1062898, F9, 23) (dual of [1062898, 1062762, 24]-code), using
- net defined by OOA [i] based on linear OOA(9136, 96627, F9, 23, 23) (dual of [(96627, 23), 2222285, 24]-NRT-code), using
(114, 114+23, 1062900)-Net over F9 — Digital
Digital (114, 137, 1062900)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9137, 1062900, F9, 23) (dual of [1062900, 1062763, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9136, 1062898, F9, 23) (dual of [1062898, 1062762, 24]-code), using
- trace code [i] based on linear OA(8168, 531449, F81, 23) (dual of [531449, 531381, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8167, 531442, F81, 23) (dual of [531442, 531375, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8161, 531442, F81, 21) (dual of [531442, 531381, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 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
- trace code [i] based on linear OA(8168, 531449, F81, 23) (dual of [531449, 531381, 24]-code), using
- linear OA(9136, 1062899, F9, 22) (dual of [1062899, 1062763, 23]-code), using Gilbert–Varšamov bound and bm = 9136 > Vbs−1(k−1) = 649807 125792 107170 646530 448605 347083 672970 807525 213033 237002 005996 685513 170245 038988 598405 148506 554229 478374 693505 713516 001745 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9136, 1062898, F9, 23) (dual of [1062898, 1062762, 24]-code), using
- construction X with Varšamov bound [i] based on
(114, 114+23, large)-Net in Base 9 — Upper bound on s
There is no (114, 137, large)-net in base 9, because
- 21 times m-reduction [i] would yield (114, 116, large)-net in base 9, but