Best Known (135−22, 135, s)-Nets in Base 9
(135−22, 135, 434816)-Net over F9 — Constructive and digital
Digital (113, 135, 434816)-net over F9, using
- 91 times duplication [i] based on digital (112, 134, 434816)-net over F9, using
- net defined by OOA [i] based on linear OOA(9134, 434816, F9, 22, 22) (dual of [(434816, 22), 9565818, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(9134, 4782976, F9, 22) (dual of [4782976, 4782842, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(9134, 4782969, F9, 22) (dual of [4782969, 4782835, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(9127, 4782969, F9, 21) (dual of [4782969, 4782842, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- OA 11-folding and stacking [i] based on linear OA(9134, 4782976, F9, 22) (dual of [4782976, 4782842, 23]-code), using
- net defined by OOA [i] based on linear OOA(9134, 434816, F9, 22, 22) (dual of [(434816, 22), 9565818, 23]-NRT-code), using
(135−22, 135, 2568261)-Net over F9 — Digital
Digital (113, 135, 2568261)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9135, 2568261, F9, 22) (dual of [2568261, 2568126, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(9135, 4782984, F9, 22) (dual of [4782984, 4782849, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(9134, 4782969, F9, 22) (dual of [4782969, 4782835, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(9120, 4782969, F9, 20) (dual of [4782969, 4782849, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(9135, 4782984, F9, 22) (dual of [4782984, 4782849, 23]-code), using
(135−22, 135, large)-Net in Base 9 — Upper bound on s
There is no (113, 135, large)-net in base 9, because
- 20 times m-reduction [i] would yield (113, 115, large)-net in base 9, but