Best Known (114−20, 114, s)-Nets in Base 9
(114−20, 114, 53160)-Net over F9 — Constructive and digital
Digital (94, 114, 53160)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 16)-net over F9, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 1 and N(F) ≥ 16, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- digital (83, 103, 53144)-net over F9, using
- net defined by OOA [i] based on linear OOA(9103, 53144, F9, 20, 20) (dual of [(53144, 20), 1062777, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(9103, 531440, F9, 20) (dual of [531440, 531337, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(9103, 531440, F9, 20) (dual of [531440, 531337, 21]-code), using
- net defined by OOA [i] based on linear OOA(9103, 53144, F9, 20, 20) (dual of [(53144, 20), 1062777, 21]-NRT-code), using
- digital (1, 11, 16)-net over F9, using
(114−20, 114, 531489)-Net over F9 — Digital
Digital (94, 114, 531489)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9114, 531489, F9, 20) (dual of [531489, 531375, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9113, 531487, F9, 20) (dual of [531487, 531374, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(12) [i] based on
- linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(967, 531441, F9, 13) (dual of [531441, 531374, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(910, 46, F9, 6) (dual of [46, 36, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(910, 52, F9, 6) (dual of [52, 42, 7]-code), using
- a “Gra†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(910, 52, F9, 6) (dual of [52, 42, 7]-code), using
- construction X applied to Ce(19) ⊂ Ce(12) [i] based on
- linear OA(9113, 531488, F9, 19) (dual of [531488, 531375, 20]-code), using Gilbert–Varšamov bound and bm = 9113 > Vbs−1(k−1) = 32 212408 278628 701342 492666 270287 826405 846605 482582 430300 423576 136139 602309 882310 282074 562570 409053 782841 [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(9113, 531487, F9, 20) (dual of [531487, 531374, 21]-code), using
- construction X with Varšamov bound [i] based on
(114−20, 114, large)-Net in Base 9 — Upper bound on s
There is no (94, 114, large)-net in base 9, because
- 18 times m-reduction [i] would yield (94, 96, large)-net in base 9, but