Best Known (48, 48+9, s)-Nets in Base 9
(48, 48+9, 1195743)-Net over F9 — Constructive and digital
Digital (48, 57, 1195743)-net over F9, using
- net defined by OOA [i] based on linear OOA(957, 1195743, F9, 9, 9) (dual of [(1195743, 9), 10761630, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(957, 4782973, F9, 9) (dual of [4782973, 4782916, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(957, 4782976, F9, 9) (dual of [4782976, 4782919, 10]-code), using
- 1 times truncation [i] based on linear OA(958, 4782977, F9, 10) (dual of [4782977, 4782919, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(950, 4782969, F9, 8) (dual of [4782969, 4782919, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(91, 8, F9, 1) (dual of [8, 7, 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(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(958, 4782977, F9, 10) (dual of [4782977, 4782919, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(957, 4782976, F9, 9) (dual of [4782976, 4782919, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(957, 4782973, F9, 9) (dual of [4782973, 4782916, 10]-code), using
(48, 48+9, 4782976)-Net over F9 — Digital
Digital (48, 57, 4782976)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(957, 4782976, F9, 9) (dual of [4782976, 4782919, 10]-code), using
- 1 times truncation [i] based on linear OA(958, 4782977, F9, 10) (dual of [4782977, 4782919, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- 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(950, 4782969, F9, 8) (dual of [4782969, 4782919, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(91, 8, F9, 1) (dual of [8, 7, 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(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(958, 4782977, F9, 10) (dual of [4782977, 4782919, 11]-code), using
(48, 48+9, large)-Net in Base 9 — Upper bound on s
There is no (48, 57, large)-net in base 9, because
- 7 times m-reduction [i] would yield (48, 50, large)-net in base 9, but