Best Known (128−20, 128, s)-Nets in Base 9
(128−20, 128, 478301)-Net over F9 — Constructive and digital
Digital (108, 128, 478301)-net over F9, using
- 91 times duplication [i] based on digital (107, 127, 478301)-net over F9, using
- net defined by OOA [i] based on linear OOA(9127, 478301, F9, 20, 20) (dual of [(478301, 20), 9565893, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(9127, 4783010, F9, 20) (dual of [4783010, 4782883, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9127, 4783011, F9, 20) (dual of [4783011, 4782884, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- 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(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(97, 42, F9, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(9127, 4783011, F9, 20) (dual of [4783011, 4782884, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(9127, 4783010, F9, 20) (dual of [4783010, 4782883, 21]-code), using
- net defined by OOA [i] based on linear OOA(9127, 478301, F9, 20, 20) (dual of [(478301, 20), 9565893, 21]-NRT-code), using
(128−20, 128, 4783013)-Net over F9 — Digital
Digital (108, 128, 4783013)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9128, 4783013, F9, 20) (dual of [4783013, 4782885, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9127, 4783011, F9, 20) (dual of [4783011, 4782884, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- 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(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(97, 42, F9, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(9127, 4783012, F9, 19) (dual of [4783012, 4782885, 20]-code), using Gilbert–Varšamov bound and bm = 9127 > Vbs−1(k−1) = 4 829363 457138 412767 875779 374620 638873 406380 280150 402133 937943 475917 632277 984210 721381 094416 030461 837954 750744 491661 902809 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 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
- linear OA(9127, 4783011, F9, 20) (dual of [4783011, 4782884, 21]-code), using
- construction X with Varšamov bound [i] based on
(128−20, 128, large)-Net in Base 9 — Upper bound on s
There is no (108, 128, large)-net in base 9, because
- 18 times m-reduction [i] would yield (108, 110, large)-net in base 9, but