Best Known (98−19, 98, s)-Nets in Base 9
(98−19, 98, 59049)-Net over F9 — Constructive and digital
Digital (79, 98, 59049)-net over F9, using
- 91 times duplication [i] based on digital (78, 97, 59049)-net over F9, using
- net defined by OOA [i] based on linear OOA(997, 59049, F9, 19, 19) (dual of [(59049, 19), 1121834, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(997, 531442, F9, 19) (dual of [531442, 531345, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(997, 531442, F9, 19) (dual of [531442, 531345, 20]-code), using
- net defined by OOA [i] based on linear OOA(997, 59049, F9, 19, 19) (dual of [(59049, 19), 1121834, 20]-NRT-code), using
(98−19, 98, 265724)-Net over F9 — Digital
Digital (79, 98, 265724)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(998, 265724, F9, 2, 19) (dual of [(265724, 2), 531350, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(998, 531448, F9, 19) (dual of [531448, 531350, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(997, 531441, F9, 19) (dual of [531441, 531344, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(991, 531441, F9, 17) (dual of [531441, 531350, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(91, 7, F9, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(998, 531448, F9, 19) (dual of [531448, 531350, 20]-code), using
(98−19, 98, large)-Net in Base 9 — Upper bound on s
There is no (79, 98, large)-net in base 9, because
- 17 times m-reduction [i] would yield (79, 81, large)-net in base 9, but