Best Known (109, 109+25, s)-Nets in Base 9
(109, 109+25, 44287)-Net over F9 — Constructive and digital
Digital (109, 134, 44287)-net over F9, using
- 91 times duplication [i] based on digital (108, 133, 44287)-net over F9, using
- net defined by OOA [i] based on linear OOA(9133, 44287, F9, 25, 25) (dual of [(44287, 25), 1107042, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9133, 531445, F9, 25) (dual of [531445, 531312, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9133, 531447, F9, 25) (dual of [531447, 531314, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(9133, 531441, F9, 25) (dual of [531441, 531308, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(9127, 531441, F9, 24) (dual of [531441, 531314, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(90, 6, F9, 0) (dual of [6, 6, 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
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(9133, 531447, F9, 25) (dual of [531447, 531314, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9133, 531445, F9, 25) (dual of [531445, 531312, 26]-code), using
- net defined by OOA [i] based on linear OOA(9133, 44287, F9, 25, 25) (dual of [(44287, 25), 1107042, 26]-NRT-code), using
(109, 109+25, 388474)-Net over F9 — Digital
Digital (109, 134, 388474)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9134, 388474, F9, 25) (dual of [388474, 388340, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9134, 531455, F9, 25) (dual of [531455, 531321, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(9133, 531442, F9, 25) (dual of [531442, 531309, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(9121, 531442, F9, 23) (dual of [531442, 531321, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 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
- discarding factors / shortening the dual code based on linear OA(9134, 531455, F9, 25) (dual of [531455, 531321, 26]-code), using
(109, 109+25, large)-Net in Base 9 — Upper bound on s
There is no (109, 134, large)-net in base 9, because
- 23 times m-reduction [i] would yield (109, 111, large)-net in base 9, but