Best Known (55, 55+18, s)-Nets in Base 9
(55, 55+18, 1459)-Net over F9 — Constructive and digital
Digital (55, 73, 1459)-net over F9, using
- 91 times duplication [i] based on digital (54, 72, 1459)-net over F9, using
- net defined by OOA [i] based on linear OOA(972, 1459, F9, 18, 18) (dual of [(1459, 18), 26190, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(972, 13131, F9, 18) (dual of [13131, 13059, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(972, 13132, F9, 18) (dual of [13132, 13060, 19]-code), using
- trace code [i] based on linear OA(8136, 6566, F81, 18) (dual of [6566, 6530, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(17) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(8136, 6566, F81, 18) (dual of [6566, 6530, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(972, 13132, F9, 18) (dual of [13132, 13060, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(972, 13131, F9, 18) (dual of [13131, 13059, 19]-code), using
- net defined by OOA [i] based on linear OOA(972, 1459, F9, 18, 18) (dual of [(1459, 18), 26190, 19]-NRT-code), using
(55, 55+18, 13134)-Net over F9 — Digital
Digital (55, 73, 13134)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(973, 13134, F9, 18) (dual of [13134, 13061, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(972, 13132, F9, 18) (dual of [13132, 13060, 19]-code), using
- trace code [i] based on linear OA(8136, 6566, F81, 18) (dual of [6566, 6530, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(17) ⊂ Ce(15) [i] based on
- trace code [i] based on linear OA(8136, 6566, F81, 18) (dual of [6566, 6530, 19]-code), using
- linear OA(972, 13133, F9, 17) (dual of [13133, 13061, 18]-code), using Gilbert–Varšamov bound and bm = 972 > Vbs−1(k−1) = 10 428221 995624 204818 030659 988479 301435 566557 242611 828952 674564 319969 [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(972, 13132, F9, 18) (dual of [13132, 13060, 19]-code), using
- construction X with Varšamov bound [i] based on
(55, 55+18, large)-Net in Base 9 — Upper bound on s
There is no (55, 73, large)-net in base 9, because
- 16 times m-reduction [i] would yield (55, 57, large)-net in base 9, but