Best Known (76, 76+25, s)-Nets in Base 9
(76, 76+25, 1094)-Net over F9 — Constructive and digital
Digital (76, 101, 1094)-net over F9, using
- 91 times duplication [i] based on digital (75, 100, 1094)-net over F9, using
- net defined by OOA [i] based on linear OOA(9100, 1094, F9, 25, 25) (dual of [(1094, 25), 27250, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9100, 13129, F9, 25) (dual of [13129, 13029, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9100, 13134, F9, 25) (dual of [13134, 13034, 26]-code), using
- trace code [i] based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 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([0,12]) ⊂ C([0,11]) [i] based on
- trace code [i] based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9100, 13134, F9, 25) (dual of [13134, 13034, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9100, 13129, F9, 25) (dual of [13129, 13029, 26]-code), using
- net defined by OOA [i] based on linear OOA(9100, 1094, F9, 25, 25) (dual of [(1094, 25), 27250, 26]-NRT-code), using
(76, 76+25, 13136)-Net over F9 — Digital
Digital (76, 101, 13136)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9101, 13136, F9, 25) (dual of [13136, 13035, 26]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9100, 13134, F9, 25) (dual of [13134, 13034, 26]-code), using
- trace code [i] based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 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([0,12]) ⊂ C([0,11]) [i] based on
- trace code [i] based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- linear OA(9100, 13135, F9, 24) (dual of [13135, 13035, 25]-code), using Gilbert–Varšamov bound and bm = 9100 > Vbs−1(k−1) = 1184 216735 762336 019216 232926 170958 733094 706304 979372 088529 823973 108738 299187 163175 838891 741489 [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(9100, 13134, F9, 25) (dual of [13134, 13034, 26]-code), using
- construction X with Varšamov bound [i] based on
(76, 76+25, large)-Net in Base 9 — Upper bound on s
There is no (76, 101, large)-net in base 9, because
- 23 times m-reduction [i] would yield (76, 78, large)-net in base 9, but