Best Known (131−12, 131, s)-Nets in Base 9
(131−12, 131, 5592400)-Net over F9 — Constructive and digital
Digital (119, 131, 5592400)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (35, 41, 2796201)-net over F9, using
- net defined by OOA [i] based on linear OOA(941, 2796201, F9, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(941, large, F9, 6) (dual of [large, large−41, 7]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(941, large, F9, 6) (dual of [large, large−41, 7]-code), using
- net defined by OOA [i] based on linear OOA(941, 2796201, F9, 6, 6) (dual of [(2796201, 6), 16777165, 7]-NRT-code), using
- digital (78, 90, 2796200)-net over F9, using
- net defined by OOA [i] based on linear OOA(990, 2796200, F9, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(990, 8388601, F9, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(990, 8388602, F9, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- trace code [i] based on linear OOA(8145, 4194301, F81, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- OOA 2-folding [i] based on linear OA(8145, 8388602, F81, 12) (dual of [8388602, 8388557, 13]-code), using
- trace code [i] based on linear OOA(8145, 4194301, F81, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(990, 8388602, F9, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(990, 8388601, F9, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(990, 2796200, F9, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- digital (35, 41, 2796201)-net over F9, using
(131−12, 131, large)-Net over F9 — Digital
Digital (119, 131, large)-net over F9, using
- t-expansion [i] based on digital (117, 131, large)-net over F9, using
- 6 times m-reduction [i] based on digital (117, 137, large)-net over F9, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9137, large, F9, 20) (dual of [large, large−137, 21]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 98−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(9137, large, F9, 20) (dual of [large, large−137, 21]-code), using
- 6 times m-reduction [i] based on digital (117, 137, large)-net over F9, using
(131−12, 131, large)-Net in Base 9 — Upper bound on s
There is no (119, 131, large)-net in base 9, because
- 10 times m-reduction [i] would yield (119, 121, large)-net in base 9, but