Best Known (83, 83+15, s)-Nets in Base 27
(83, 83+15, 1375522)-Net over F27 — Constructive and digital
Digital (83, 98, 1375522)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (20, 27, 177151)-net over F27, using
- net defined by OOA [i] based on linear OOA(2727, 177151, F27, 7, 7) (dual of [(177151, 7), 1240030, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2727, 531454, F27, 7) (dual of [531454, 531427, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2727, 531455, F27, 7) (dual of [531455, 531428, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(2725, 531441, F27, 7) (dual of [531441, 531416, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2713, 531441, F27, 4) (dual of [531441, 531428, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(2727, 531455, F27, 7) (dual of [531455, 531428, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2727, 531454, F27, 7) (dual of [531454, 531427, 8]-code), using
- net defined by OOA [i] based on linear OOA(2727, 177151, F27, 7, 7) (dual of [(177151, 7), 1240030, 8]-NRT-code), using
- digital (56, 71, 1198371)-net over F27, using
- net defined by OOA [i] based on linear OOA(2771, 1198371, F27, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2771, 8388598, F27, 15) (dual of [8388598, 8388527, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2771, large, F27, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2771, large, F27, 15) (dual of [large, large−71, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2771, 8388598, F27, 15) (dual of [8388598, 8388527, 16]-code), using
- net defined by OOA [i] based on linear OOA(2771, 1198371, F27, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- digital (20, 27, 177151)-net over F27, using
(83, 83+15, large)-Net over F27 — Digital
Digital (83, 98, large)-net over F27, using
- t-expansion [i] based on digital (80, 98, large)-net over F27, using
- 3 times m-reduction [i] based on digital (80, 101, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- 3 times m-reduction [i] based on digital (80, 101, large)-net over F27, using
(83, 83+15, large)-Net in Base 27 — Upper bound on s
There is no (83, 98, large)-net in base 27, because
- 13 times m-reduction [i] would yield (83, 85, large)-net in base 27, but