Best Known (52, 64, s)-Nets in Base 9
(52, 64, 88576)-Net over F9 — Constructive and digital
Digital (52, 64, 88576)-net over F9, using
- 91 times duplication [i] based on digital (51, 63, 88576)-net over F9, using
- net defined by OOA [i] based on linear OOA(963, 88576, F9, 12, 12) (dual of [(88576, 12), 1062849, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(963, 531456, F9, 12) (dual of [531456, 531393, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(962, 531455, F9, 12) (dual of [531455, 531393, 13]-code), using
- construction X4 applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(961, 531441, F9, 12) (dual of [531441, 531380, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(913, 14, F9, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,9)), using
- dual of repetition code with length 14 [i]
- linear OA(91, 14, F9, 1) (dual of [14, 13, 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
- construction X4 applied to Ce(11) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(962, 531455, F9, 12) (dual of [531455, 531393, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(963, 531456, F9, 12) (dual of [531456, 531393, 13]-code), using
- net defined by OOA [i] based on linear OOA(963, 88576, F9, 12, 12) (dual of [(88576, 12), 1062849, 13]-NRT-code), using
(52, 64, 531458)-Net over F9 — Digital
Digital (52, 64, 531458)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(964, 531458, F9, 12) (dual of [531458, 531394, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(962, 531455, F9, 12) (dual of [531455, 531393, 13]-code), using
- construction X4 applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(961, 531441, F9, 12) (dual of [531441, 531380, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(913, 14, F9, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,9)), using
- dual of repetition code with length 14 [i]
- linear OA(91, 14, F9, 1) (dual of [14, 13, 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
- construction X4 applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(962, 531456, F9, 10) (dual of [531456, 531394, 11]-code), using Gilbert–Varšamov bound and bm = 962 > Vbs−1(k−1) = 1 250883 392309 063443 963462 891145 859019 664879 904729 984569 [i]
- linear OA(91, 2, F9, 1) (dual of [2, 1, 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
- linear OA(962, 531455, F9, 12) (dual of [531455, 531393, 13]-code), using
- construction X with Varšamov bound [i] based on
(52, 64, large)-Net in Base 9 — Upper bound on s
There is no (52, 64, large)-net in base 9, because
- 10 times m-reduction [i] would yield (52, 54, large)-net in base 9, but