Best Known (113, 113+31, s)-Nets in Base 9
(113, 113+31, 3938)-Net over F9 — Constructive and digital
Digital (113, 144, 3938)-net over F9, using
- 92 times duplication [i] based on digital (111, 142, 3938)-net over F9, using
- net defined by OOA [i] based on linear OOA(9142, 3938, F9, 31, 31) (dual of [(3938, 31), 121936, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(9142, 59071, F9, 31) (dual of [59071, 58929, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(9142, 59075, F9, 31) (dual of [59075, 58933, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(9136, 59049, F9, 31) (dual of [59049, 58913, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(9116, 59049, F9, 26) (dual of [59049, 58933, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(96, 26, F9, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(9142, 59075, F9, 31) (dual of [59075, 58933, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(9142, 59071, F9, 31) (dual of [59071, 58929, 32]-code), using
- net defined by OOA [i] based on linear OOA(9142, 3938, F9, 31, 31) (dual of [(3938, 31), 121936, 32]-NRT-code), using
(113, 113+31, 59083)-Net over F9 — Digital
Digital (113, 144, 59083)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9144, 59083, F9, 31) (dual of [59083, 58939, 32]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9143, 59081, F9, 31) (dual of [59081, 58938, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(9136, 59049, F9, 31) (dual of [59049, 58913, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(9111, 59049, F9, 25) (dual of [59049, 58938, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(97, 32, F9, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [i] based on
- linear OA(9143, 59082, F9, 30) (dual of [59082, 58939, 31]-code), using Gilbert–Varšamov bound and bm = 9143 > Vbs−1(k−1) = 40 935474 945383 310002 397984 639599 031806 394981 854493 529621 195622 042944 633986 384740 158498 823430 734310 111359 087917 914915 861979 741799 178057 [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(9143, 59081, F9, 31) (dual of [59081, 58938, 32]-code), using
- construction X with Varšamov bound [i] based on
(113, 113+31, large)-Net in Base 9 — Upper bound on s
There is no (113, 144, large)-net in base 9, because
- 29 times m-reduction [i] would yield (113, 115, large)-net in base 9, but