Best Known (32, 32+9, s)-Nets in Base 9
(32, 32+9, 14763)-Net over F9 — Constructive and digital
Digital (32, 41, 14763)-net over F9, using
- net defined by OOA [i] based on linear OOA(941, 14763, F9, 9, 9) (dual of [(14763, 9), 132826, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(941, 59053, F9, 9) (dual of [59053, 59012, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(941, 59054, F9, 9) (dual of [59054, 59013, 10]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(942, 59055, F9, 10) (dual of [59055, 59013, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(941, 59054, F9, 9) (dual of [59054, 59013, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(941, 59053, F9, 9) (dual of [59053, 59012, 10]-code), using
(32, 32+9, 59054)-Net over F9 — Digital
Digital (32, 41, 59054)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(941, 59054, F9, 9) (dual of [59054, 59013, 10]-code), using
- 1 times truncation [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
- 1 times truncation [i] based on linear OA(942, 59055, F9, 10) (dual of [59055, 59013, 11]-code), using
(32, 32+9, large)-Net in Base 9 — Upper bound on s
There is no (32, 41, large)-net in base 9, because
- 7 times m-reduction [i] would yield (32, 34, large)-net in base 9, but