Best Known (137−21, 137, s)-Nets in Base 9
(137−21, 137, 478307)-Net over F9 — Constructive and digital
Digital (116, 137, 478307)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 10, 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 (106, 127, 478297)-net over F9, using
- net defined by OOA [i] based on linear OOA(9127, 478297, F9, 21, 21) (dual of [(478297, 21), 10044110, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9127, 4782971, F9, 21) (dual of [4782971, 4782844, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9127, 4782976, F9, 21) (dual of [4782976, 4782849, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- 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(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(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(20) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(9127, 4782976, F9, 21) (dual of [4782976, 4782849, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9127, 4782971, F9, 21) (dual of [4782971, 4782844, 22]-code), using
- net defined by OOA [i] based on linear OOA(9127, 478297, F9, 21, 21) (dual of [(478297, 21), 10044110, 22]-NRT-code), using
- digital (0, 10, 10)-net over F9, using
(137−21, 137, 4783021)-Net over F9 — Digital
Digital (116, 137, 4783021)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9137, 4783021, F9, 21) (dual of [4783021, 4782884, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- 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(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(910, 52, F9, 6) (dual of [52, 42, 7]-code), using
- a “Gra†code from Grassl’s database [i]
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
(137−21, 137, large)-Net in Base 9 — Upper bound on s
There is no (116, 137, large)-net in base 9, because
- 19 times m-reduction [i] would yield (116, 118, large)-net in base 9, but