Best Known (73, 73+13, s)-Nets in Base 9
(73, 73+13, 797182)-Net over F9 — Constructive and digital
Digital (73, 86, 797182)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 20)-net over F9, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 2 and N(F) ≥ 20, using
- net from sequence [i] based on digital (2, 19)-sequence over F9, using
- digital (65, 78, 797162)-net over F9, using
- net defined by OOA [i] based on linear OOA(978, 797162, F9, 13, 13) (dual of [(797162, 13), 10363028, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(978, 4782973, F9, 13) (dual of [4782973, 4782895, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(978, 4782976, F9, 13) (dual of [4782976, 4782898, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(971, 4782969, F9, 12) (dual of [4782969, 4782898, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(978, 4782976, F9, 13) (dual of [4782976, 4782898, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(978, 4782973, F9, 13) (dual of [4782973, 4782895, 14]-code), using
- net defined by OOA [i] based on linear OOA(978, 797162, F9, 13, 13) (dual of [(797162, 13), 10363028, 14]-NRT-code), using
- digital (2, 8, 20)-net over F9, using
(73, 73+13, 4783012)-Net over F9 — Digital
Digital (73, 86, 4783012)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(986, 4783012, F9, 13) (dual of [4783012, 4782926, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(985, 4783011, F9, 13) (dual of [4783011, 4782926, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(943, 4782969, F9, 7) (dual of [4782969, 4782926, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(97, 42, F9, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(985, 4783011, F9, 13) (dual of [4783011, 4782926, 14]-code), using
(73, 73+13, large)-Net in Base 9 — Upper bound on s
There is no (73, 86, large)-net in base 9, because
- 11 times m-reduction [i] would yield (73, 75, large)-net in base 9, but