Best Known (79, 79+26, s)-Nets in Base 9
(79, 79+26, 1010)-Net over F9 — Constructive and digital
Digital (79, 105, 1010)-net over F9, using
- 91 times duplication [i] based on digital (78, 104, 1010)-net over F9, using
- net defined by OOA [i] based on linear OOA(9104, 1010, F9, 26, 26) (dual of [(1010, 26), 26156, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(9104, 13130, F9, 26) (dual of [13130, 13026, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(9104, 13132, F9, 26) (dual of [13132, 13028, 27]-code), using
- trace code [i] based on linear OA(8152, 6566, F81, 26) (dual of [6566, 6514, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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 Ce(25) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(8152, 6566, F81, 26) (dual of [6566, 6514, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(9104, 13132, F9, 26) (dual of [13132, 13028, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(9104, 13130, F9, 26) (dual of [13130, 13026, 27]-code), using
- net defined by OOA [i] based on linear OOA(9104, 1010, F9, 26, 26) (dual of [(1010, 26), 26156, 27]-NRT-code), using
(79, 79+26, 13134)-Net over F9 — Digital
Digital (79, 105, 13134)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9105, 13134, F9, 26) (dual of [13134, 13029, 27]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9104, 13132, F9, 26) (dual of [13132, 13028, 27]-code), using
- trace code [i] based on linear OA(8152, 6566, F81, 26) (dual of [6566, 6514, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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 Ce(25) ⊂ Ce(23) [i] based on
- trace code [i] based on linear OA(8152, 6566, F81, 26) (dual of [6566, 6514, 27]-code), using
- linear OA(9104, 13133, F9, 25) (dual of [13133, 13029, 26]-code), using Gilbert–Varšamov bound and bm = 9104 > Vbs−1(k−1) = 5 156573 588554 532887 740450 066001 614856 140581 340273 089180 466540 468028 761346 162016 642401 768011 999969 [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(9104, 13132, F9, 26) (dual of [13132, 13028, 27]-code), using
- construction X with Varšamov bound [i] based on
(79, 79+26, large)-Net in Base 9 — Upper bound on s
There is no (79, 105, large)-net in base 9, because
- 24 times m-reduction [i] would yield (79, 81, large)-net in base 9, but