Best Known (80−13, 80, s)-Nets in Base 9
(80−13, 80, 797164)-Net over F9 — Constructive and digital
Digital (67, 80, 797164)-net over F9, using
- 91 times duplication [i] based on digital (66, 79, 797164)-net over F9, using
- net defined by OOA [i] based on linear OOA(979, 797164, F9, 13, 13) (dual of [(797164, 13), 10363053, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(979, 4782985, F9, 13) (dual of [4782985, 4782906, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [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(964, 4782969, F9, 11) (dual of [4782969, 4782905, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(915, 16, F9, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,9)), using
- dual of repetition code with length 16 [i]
- linear OA(91, 16, F9, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(979, 4782985, F9, 13) (dual of [4782985, 4782906, 14]-code), using
- net defined by OOA [i] based on linear OOA(979, 797164, F9, 13, 13) (dual of [(797164, 13), 10363053, 14]-NRT-code), using
(80−13, 80, 4376428)-Net over F9 — Digital
Digital (67, 80, 4376428)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(980, 4376428, F9, 13) (dual of [4376428, 4376348, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(980, 4782986, F9, 13) (dual of [4782986, 4782906, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(979, 4782985, F9, 13) (dual of [4782985, 4782906, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [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(964, 4782969, F9, 11) (dual of [4782969, 4782905, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(915, 16, F9, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,9)), using
- dual of repetition code with length 16 [i]
- linear OA(91, 16, F9, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(979, 4782985, F9, 13) (dual of [4782985, 4782906, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(980, 4782986, F9, 13) (dual of [4782986, 4782906, 14]-code), using
(80−13, 80, large)-Net in Base 9 — Upper bound on s
There is no (67, 80, large)-net in base 9, because
- 11 times m-reduction [i] would yield (67, 69, large)-net in base 9, but