Best Known (59, 72, s)-Nets in Base 9
(59, 72, 88577)-Net over F9 — Constructive and digital
Digital (59, 72, 88577)-net over F9, using
- 91 times duplication [i] based on digital (58, 71, 88577)-net over F9, using
- net defined by OOA [i] based on linear OOA(971, 88577, F9, 13, 13) (dual of [(88577, 13), 1151430, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(971, 531463, F9, 13) (dual of [531463, 531392, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(970, 531462, F9, 13) (dual of [531462, 531392, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(967, 531441, F9, 13) (dual of [531441, 531374, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(93, 21, F9, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(970, 531462, F9, 13) (dual of [531462, 531392, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(971, 531463, F9, 13) (dual of [531463, 531392, 14]-code), using
- net defined by OOA [i] based on linear OOA(971, 88577, F9, 13, 13) (dual of [(88577, 13), 1151430, 14]-NRT-code), using
(59, 72, 531466)-Net over F9 — Digital
Digital (59, 72, 531466)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(972, 531466, F9, 13) (dual of [531466, 531394, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(970, 531462, F9, 13) (dual of [531462, 531392, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(967, 531441, F9, 13) (dual of [531441, 531374, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(93, 21, F9, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(970, 531464, F9, 12) (dual of [531464, 531394, 13]-code), using Gilbert–Varšamov bound and bm = 970 > Vbs−1(k−1) = 205586 077138 049805 229100 691626 015791 635194 115897 049997 412325 874553 [i]
- linear OA(90, 2, F9, 0) (dual of [2, 2, 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(970, 531462, F9, 13) (dual of [531462, 531392, 14]-code), using
- construction X with Varšamov bound [i] based on
(59, 72, large)-Net in Base 9 — Upper bound on s
There is no (59, 72, large)-net in base 9, because
- 11 times m-reduction [i] would yield (59, 61, large)-net in base 9, but