Best Known (61, 61+12, s)-Nets in Base 9
(61, 61+12, 797164)-Net over F9 — Constructive and digital
Digital (61, 73, 797164)-net over F9, using
- 91 times duplication [i] based on digital (60, 72, 797164)-net over F9, using
- net defined by OOA [i] based on linear OOA(972, 797164, F9, 12, 12) (dual of [(797164, 12), 9565896, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(972, 4782984, F9, 12) (dual of [4782984, 4782912, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- 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(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(91, 15, F9, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OA 6-folding and stacking [i] based on linear OA(972, 4782984, F9, 12) (dual of [4782984, 4782912, 13]-code), using
- net defined by OOA [i] based on linear OOA(972, 797164, F9, 12, 12) (dual of [(797164, 12), 9565896, 13]-NRT-code), using
(61, 61+12, 4201766)-Net over F9 — Digital
Digital (61, 73, 4201766)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(973, 4201766, F9, 12) (dual of [4201766, 4201693, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(973, 4782986, F9, 12) (dual of [4782986, 4782913, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(972, 4782985, F9, 12) (dual of [4782985, 4782913, 13]-code), using
- construction X4 applied to Ce(11) ⊂ Ce(9) [i] based on
- 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(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(11) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(972, 4782985, F9, 12) (dual of [4782985, 4782913, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(973, 4782986, F9, 12) (dual of [4782986, 4782913, 13]-code), using
(61, 61+12, large)-Net in Base 9 — Upper bound on s
There is no (61, 73, large)-net in base 9, because
- 10 times m-reduction [i] would yield (61, 63, large)-net in base 9, but