Best Known (125, 125+13, s)-Nets in Base 9
(125, 125+13, 4390607)-Net over F9 — Constructive and digital
Digital (125, 138, 4390607)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (34, 40, 1594407)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 82)-net over F9, using
- net defined by OOA [i] based on linear OOA(94, 82, F9, 3, 3) (dual of [(82, 3), 242, 4]-NRT-code), using
- digital (30, 36, 1594325)-net over F9, using
- net defined by OOA [i] based on linear OOA(936, 1594325, F9, 6, 6) (dual of [(1594325, 6), 9565914, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(936, 4782975, F9, 6) (dual of [4782975, 4782939, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(936, 4782976, F9, 6) (dual of [4782976, 4782940, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(936, 4782969, F9, 6) (dual of [4782969, 4782933, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(929, 4782969, F9, 5) (dual of [4782969, 4782940, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(936, 4782976, F9, 6) (dual of [4782976, 4782940, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(936, 4782975, F9, 6) (dual of [4782975, 4782939, 7]-code), using
- net defined by OOA [i] based on linear OOA(936, 1594325, F9, 6, 6) (dual of [(1594325, 6), 9565914, 7]-NRT-code), using
- digital (1, 4, 82)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (85, 98, 2796200)-net over F9, using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- trace code [i] based on linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- trace code [i] based on linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- digital (34, 40, 1594407)-net over F9, using
(125, 125+13, 5592400)-Net in Base 9 — Constructive
(125, 138, 5592400)-net in base 9, using
- 91 times duplication [i] based on (124, 137, 5592400)-net in base 9, using
- (u, u+v)-construction [i] based on
- (33, 39, 2796201)-net in base 9, using
- base change [i] based on digital (20, 26, 2796201)-net over F27, using
- net defined by OOA [i] based on linear OOA(2726, 2796201, F27, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(2726, large, F27, 6) (dual of [large, large−26, 7]-code), using
- net defined by OOA [i] based on linear OOA(2726, 2796201, F27, 6, 6) (dual of [(2796201, 6), 16777180, 7]-NRT-code), using
- base change [i] based on digital (20, 26, 2796201)-net over F27, using
- digital (85, 98, 2796200)-net over F9, using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- trace code [i] based on linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8149, large, F81, 13) (dual of [large, large−49, 14]-code), using
- OOA 2-folding [i] based on linear OA(8149, 8388602, F81, 13) (dual of [8388602, 8388553, 14]-code), using
- trace code [i] based on linear OOA(8149, 4194301, F81, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(998, 8388602, F9, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(998, 8388601, F9, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(998, 2796200, F9, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- (33, 39, 2796201)-net in base 9, using
- (u, u+v)-construction [i] based on
(125, 125+13, large)-Net over F9 — Digital
Digital (125, 138, large)-net over F9, using
- t-expansion [i] based on digital (124, 138, large)-net over F9, using
- 7 times m-reduction [i] based on digital (124, 145, large)-net over F9, using
(125, 125+13, large)-Net in Base 9 — Upper bound on s
There is no (125, 138, large)-net in base 9, because
- 11 times m-reduction [i] would yield (125, 127, large)-net in base 9, but