Best Known (52, 52+18, s)-Nets in Base 27
(52, 52+18, 59050)-Net over F27 — Constructive and digital
Digital (52, 70, 59050)-net over F27, using
- net defined by OOA [i] based on linear OOA(2770, 59050, F27, 18, 18) (dual of [(59050, 18), 1062830, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2770, 531450, F27, 18) (dual of [531450, 531380, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2761, 531441, F27, 16) (dual of [531441, 531380, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(271, 9, F27, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- OA 9-folding and stacking [i] based on linear OA(2770, 531450, F27, 18) (dual of [531450, 531380, 19]-code), using
(52, 52+18, 389326)-Net over F27 — Digital
Digital (52, 70, 389326)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2770, 389326, F27, 18) (dual of [389326, 389256, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2770, 531450, F27, 18) (dual of [531450, 531380, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(2769, 531441, F27, 18) (dual of [531441, 531372, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2761, 531441, F27, 16) (dual of [531441, 531380, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(271, 9, F27, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2770, 531450, F27, 18) (dual of [531450, 531380, 19]-code), using
(52, 52+18, large)-Net in Base 27 — Upper bound on s
There is no (52, 70, large)-net in base 27, because
- 16 times m-reduction [i] would yield (52, 54, large)-net in base 27, but