Best Known (71−18, 71, s)-Nets in Base 27
(71−18, 71, 59050)-Net over F27 — Constructive and digital
Digital (53, 71, 59050)-net over F27, using
- 271 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(2770, 59050, F27, 18, 18) (dual of [(59050, 18), 1062830, 19]-NRT-code), using
(71−18, 71, 478382)-Net over F27 — Digital
Digital (53, 71, 478382)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2771, 478382, F27, 18) (dual of [478382, 478311, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2771, 531455, F27, 18) (dual of [531455, 531384, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [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(2757, 531441, F27, 15) (dual of [531441, 531384, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(272, 14, F27, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2771, 531455, F27, 18) (dual of [531455, 531384, 19]-code), using
(71−18, 71, large)-Net in Base 27 — Upper bound on s
There is no (53, 71, large)-net in base 27, because
- 16 times m-reduction [i] would yield (53, 55, large)-net in base 27, but