Best Known (103, 103+19, s)-Nets in Base 9
(103, 103+19, 531451)-Net over F9 — Constructive and digital
Digital (103, 122, 531451)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 10)-net over F9, using
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 0 and N(F) ≥ 10, using
- the rational function field F9(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- digital (94, 113, 531441)-net over F9, using
- net defined by OOA [i] based on linear OOA(9113, 531441, F9, 19, 19) (dual of [(531441, 19), 10097266, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on 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]
- OOA 9-folding and stacking with additional row [i] based on linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using
- net defined by OOA [i] based on linear OOA(9113, 531441, F9, 19, 19) (dual of [(531441, 19), 10097266, 20]-NRT-code), using
- digital (0, 9, 10)-net over F9, using
(103, 103+19, 4783015)-Net over F9 — Digital
Digital (103, 122, 4783015)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9122, 4783015, F9, 19) (dual of [4783015, 4782893, 20]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9120, 4783011, F9, 19) (dual of [4783011, 4782891, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(12) [i] based on
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(18) ⊂ Ce(12) [i] based on
- linear OA(9120, 4783013, F9, 18) (dual of [4783013, 4782893, 19]-code), using Gilbert–Varšamov bound and bm = 9120 > Vbs−1(k−1) = 2 271820 979786 716051 824536 673470 408417 084553 375315 031398 533400 827447 232085 368185 227728 375755 193646 017289 000316 101281 [i]
- linear OA(90, 2, F9, 0) (dual of [2, 2, 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(9120, 4783011, F9, 19) (dual of [4783011, 4782891, 20]-code), using
- construction X with Varšamov bound [i] based on
(103, 103+19, large)-Net in Base 9 — Upper bound on s
There is no (103, 122, large)-net in base 9, because
- 17 times m-reduction [i] would yield (103, 105, large)-net in base 9, but