Best Known (42, 54, s)-Nets in Base 9
(42, 54, 9843)-Net over F9 — Constructive and digital
Digital (42, 54, 9843)-net over F9, using
- 92 times duplication [i] based on digital (40, 52, 9843)-net over F9, using
- net defined by OOA [i] based on linear OOA(952, 9843, F9, 12, 12) (dual of [(9843, 12), 118064, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(952, 59058, F9, 12) (dual of [59058, 59006, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(952, 59060, F9, 12) (dual of [59060, 59008, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 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(91, 11, F9, 1) (dual of [11, 10, 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(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(952, 59060, F9, 12) (dual of [59060, 59008, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(952, 59058, F9, 12) (dual of [59058, 59006, 13]-code), using
- net defined by OOA [i] based on linear OOA(952, 9843, F9, 12, 12) (dual of [(9843, 12), 118064, 13]-NRT-code), using
(42, 54, 59063)-Net over F9 — Digital
Digital (42, 54, 59063)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(954, 59063, F9, 12) (dual of [59063, 59009, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- 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, 12, F9, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(91, 2, F9, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(7) [i] based on
(42, 54, large)-Net in Base 9 — Upper bound on s
There is no (42, 54, large)-net in base 9, because
- 10 times m-reduction [i] would yield (42, 44, large)-net in base 9, but