Best Known (57, 57+11, s)-Nets in Base 9
(57, 57+11, 956598)-Net over F9 — Constructive and digital
Digital (57, 68, 956598)-net over F9, using
- net defined by OOA [i] based on linear OOA(968, 956598, F9, 11, 11) (dual of [(956598, 11), 10522510, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(968, 4782991, F9, 11) (dual of [4782991, 4782923, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(968, 4782994, F9, 11) (dual of [4782994, 4782926, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- 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(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(94, 25, F9, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,9)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(968, 4782994, F9, 11) (dual of [4782994, 4782926, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(968, 4782991, F9, 11) (dual of [4782991, 4782923, 12]-code), using
(57, 57+11, 4782994)-Net over F9 — Digital
Digital (57, 68, 4782994)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(968, 4782994, F9, 11) (dual of [4782994, 4782926, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- 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(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(94, 25, F9, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,9)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
(57, 57+11, large)-Net in Base 9 — Upper bound on s
There is no (57, 68, large)-net in base 9, because
- 9 times m-reduction [i] would yield (57, 59, large)-net in base 9, but