Best Known (128−21, 128, s)-Nets in Base 9
(128−21, 128, 478298)-Net over F9 — Constructive and digital
Digital (107, 128, 478298)-net over F9, using
- net defined by OOA [i] based on linear OOA(9128, 478298, F9, 21, 21) (dual of [(478298, 21), 10044130, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9128, 4782981, F9, 21) (dual of [4782981, 4782853, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9128, 4782985, F9, 21) (dual of [4782985, 4782857, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(9127, 4782970, F9, 21) (dual of [4782970, 4782843, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9128, 4782985, F9, 21) (dual of [4782985, 4782857, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9128, 4782981, F9, 21) (dual of [4782981, 4782853, 22]-code), using
(128−21, 128, 2391492)-Net over F9 — Digital
Digital (107, 128, 2391492)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(9128, 2391492, F9, 2, 21) (dual of [(2391492, 2), 4782856, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(9128, 4782984, F9, 21) (dual of [4782984, 4782856, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9128, 4782985, F9, 21) (dual of [4782985, 4782857, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(9127, 4782970, F9, 21) (dual of [4782970, 4782843, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(9128, 4782985, F9, 21) (dual of [4782985, 4782857, 22]-code), using
- OOA 2-folding [i] based on linear OA(9128, 4782984, F9, 21) (dual of [4782984, 4782856, 22]-code), using
(128−21, 128, large)-Net in Base 9 — Upper bound on s
There is no (107, 128, large)-net in base 9, because
- 19 times m-reduction [i] would yield (107, 109, large)-net in base 9, but