Best Known (123, 145, s)-Nets in Base 9
(123, 145, 434826)-Net over F9 — Constructive and digital
Digital (123, 145, 434826)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 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 (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
- digital (0, 11, 10)-net over F9, using
(123, 145, 4783023)-Net over F9 — Digital
Digital (123, 145, 4783023)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9145, 4783023, F9, 22) (dual of [4783023, 4782878, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9144, 4783021, F9, 22) (dual of [4783021, 4782877, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(14) [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(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(21) ⊂ Ce(14) [i] based on
- linear OA(9144, 4783022, F9, 21) (dual of [4783022, 4782878, 22]-code), using Gilbert–Varšamov bound and bm = 9144 > Vbs−1(k−1) = 18 608177 980571 879827 221674 954233 297066 333156 919736 490850 021034 723850 720037 318690 111813 017805 434468 482909 554698 572333 355030 014677 537001 [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(9144, 4783021, F9, 22) (dual of [4783021, 4782877, 23]-code), using
- construction X with Varšamov bound [i] based on
(123, 145, large)-Net in Base 9 — Upper bound on s
There is no (123, 145, large)-net in base 9, because
- 20 times m-reduction [i] would yield (123, 125, large)-net in base 9, but