Best Known (42−10, 42, s)-Nets in Base 9
(42−10, 42, 11811)-Net over F9 — Constructive and digital
Digital (32, 42, 11811)-net over F9, using
- net defined by OOA [i] based on linear OOA(942, 11811, F9, 10, 10) (dual of [(11811, 10), 118068, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(942, 59055, F9, 10) (dual of [59055, 59013, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(936, 59049, F9, 8) (dual of [59049, 59013, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(942, 59055, F9, 10) (dual of [59055, 59013, 11]-code), using
(42−10, 42, 36563)-Net over F9 — Digital
Digital (32, 42, 36563)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(942, 36563, F9, 10) (dual of [36563, 36521, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(942, 59055, F9, 10) (dual of [59055, 59013, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(936, 59049, F9, 8) (dual of [59049, 59013, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(942, 59055, F9, 10) (dual of [59055, 59013, 11]-code), using
(42−10, 42, large)-Net in Base 9 — Upper bound on s
There is no (32, 42, large)-net in base 9, because
- 8 times m-reduction [i] would yield (32, 34, large)-net in base 9, but