Best Known (63, 81, s)-Nets in Base 9
(63, 81, 6561)-Net over F9 — Constructive and digital
Digital (63, 81, 6561)-net over F9, using
- t-expansion [i] based on digital (62, 81, 6561)-net over F9, using
- net defined by OOA [i] based on linear OOA(981, 6561, F9, 19, 19) (dual of [(6561, 19), 124578, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(981, 59050, F9, 19) (dual of [59050, 58969, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, 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(981, 59050, F9, 19) (dual of [59050, 58969, 20]-code), using
- net defined by OOA [i] based on linear OOA(981, 6561, F9, 19, 19) (dual of [(6561, 19), 124578, 20]-NRT-code), using
(63, 81, 50186)-Net over F9 — Digital
Digital (63, 81, 50186)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(981, 50186, F9, 18) (dual of [50186, 50105, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(981, 59054, F9, 18) (dual of [59054, 58973, 19]-code), using
- 1 times truncation [i] based on linear OA(982, 59055, F9, 19) (dual of [59055, 58973, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(981, 59049, F9, 19) (dual of [59049, 58968, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(976, 59049, F9, 17) (dual of [59049, 58973, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(91, 6, F9, 1) (dual of [6, 5, 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
- 1 times truncation [i] based on linear OA(982, 59055, F9, 19) (dual of [59055, 58973, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(981, 59054, F9, 18) (dual of [59054, 58973, 19]-code), using
(63, 81, large)-Net in Base 9 — Upper bound on s
There is no (63, 81, large)-net in base 9, because
- 16 times m-reduction [i] would yield (63, 65, large)-net in base 9, but