Best Known (56, 56+13, s)-Nets in Base 9
(56, 56+13, 88575)-Net over F9 — Constructive and digital
Digital (56, 69, 88575)-net over F9, using
- 91 times duplication [i] based on digital (55, 68, 88575)-net over F9, using
- net defined by OOA [i] based on linear OOA(968, 88575, F9, 13, 13) (dual of [(88575, 13), 1151407, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(968, 531451, F9, 13) (dual of [531451, 531383, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(968, 531454, F9, 13) (dual of [531454, 531386, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [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(955, 531441, F9, 11) (dual of [531441, 531386, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(968, 531454, F9, 13) (dual of [531454, 531386, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(968, 531451, F9, 13) (dual of [531451, 531383, 14]-code), using
- net defined by OOA [i] based on linear OOA(968, 88575, F9, 13, 13) (dual of [(88575, 13), 1151407, 14]-NRT-code), using
(56, 56+13, 486265)-Net over F9 — Digital
Digital (56, 69, 486265)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(969, 486265, F9, 13) (dual of [486265, 486196, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(969, 531456, F9, 13) (dual of [531456, 531387, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(968, 531455, F9, 13) (dual of [531455, 531387, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [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(955, 531441, F9, 11) (dual of [531441, 531386, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(913, 14, F9, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,9)), using
- dual of repetition code with length 14 [i]
- linear OA(91, 14, F9, 1) (dual of [14, 13, 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(968, 531455, F9, 13) (dual of [531455, 531387, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(969, 531456, F9, 13) (dual of [531456, 531387, 14]-code), using
(56, 56+13, large)-Net in Base 9 — Upper bound on s
There is no (56, 69, large)-net in base 9, because
- 11 times m-reduction [i] would yield (56, 58, large)-net in base 9, but